Interpreting Functions - Common Core: High School - Functions
Card 1 of 672
Given the graph above of 
, which intervals represent where 
is increasing?
Given the graph above of
Tap to see back →
This question is testing one's ability to grasp the relationship between the image a function creates graphically and the intervals where the function is increase, decreasing, positive, or negative. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for where the function is increasing. It is important to understand that when a function is increasing, the graph exhibits a positive slope.
Step 2: Identify the intervals where the graph is positive (increasing) and where it is negative (decreasing).
Looking at the above graph, there are two intervals where the graph is increasing and one interval where it is decreasing.
As the graph approaches 
from the left, the 
values increase. This means that the slope for this section of the graph is positive or increasing; therefore it is one of the intervals where the function 
is increasing.
Between the 
values of 
the 
values decrease. This means that the slope for this section of the graph is negative or decreasing.
From the 
value 
to infinity, the 
values increase. This means that the slope for this section of the graph is also positive or increasing; therefore it is another one of the intervals where the function 
is increasing.
Step 3: Answer the question.
is increasing on the intervals 
.
This question is testing one's ability to grasp the relationship between the image a function creates graphically and the intervals where the function is increase, decreasing, positive, or negative. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for where the function is increasing. It is important to understand that when a function is increasing, the graph exhibits a positive slope.
Step 2: Identify the intervals where the graph is positive (increasing) and where it is negative (decreasing).
Looking at the above graph, there are two intervals where the graph is increasing and one interval where it is decreasing.
As the graph approaches
Between the
From the
Step 3: Answer the question.
Given the function 
, find the 
and 
intercepts.
Given the function
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.Math.content.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the and intercepts of the function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph intersects the 
-axis and where it intersects the 
-axis as these correspond to the intercepts.
II. Algebraically, find the intercepts by substituting in zero for 
and solving for 
(calculating the 
-intercept) and then substituting in zero for 
and solving for 
(calculating the 
-intercept).
Step 3: Choose an approach from Step 2 and perform the necessary actions.
For the purpose of this question let's solve using option II.
To algebraically solve for the 
-intercept substitute zero in for 
and solve for 
. (Recall that 
)
Perform algebraic operations to combine like terms.
Isolate the variable on one side of the equation by using the opposite operation to move all other constants to the other side.
To algebraically solve for the 
-intercept substitute zero in for 
and solve for 
. (Recall that 
)
Step 4: Answer question.
Using the algebraic approach, the and intercepts were found to be as follows.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.Math.content.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph intersects the
II. Algebraically, find the intercepts by substituting in zero for
Step 3: Choose an approach from Step 2 and perform the necessary actions.
For the purpose of this question let's solve using option II.
To algebraically solve for the
Perform algebraic operations to combine like terms.
Isolate the variable on one side of the equation by using the opposite operation to move all other constants to the other side.
To algebraically solve for the
Step 4: Answer question.
Using the algebraic approach, the
What is the relative maximum of the following function?
What is the relative maximum of the following function?
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the relative maximum of the function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph reaches its peak.
Step 3: For the purpose of this question let's solve using graphing calculator technology.
Graphing the function results in the following graph.
Step 4: Trace the function using the graphing calculator to find the coordinates of the peak.
Step 5: Answer the question.
The relative maximum of the function occurs at 
.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the relative maximum of the function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph reaches its peak.
Step 3: For the purpose of this question let's solve using graphing calculator technology.
Graphing the function results in the following graph.
Step 4: Trace the function using the graphing calculator to find the coordinates of the peak.
Step 5: Answer the question.
The relative maximum of the function occurs at
Given the function 
, find the line of symmetry.
Given the function
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the axis of symmetry for this particular function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that is equal to the 
value of the vertex of the function.
Step 3: Graph the parabola and plot the vertical line that is equal to the 
value of the vertex.
Step 4: Answer the question.
The vertex is located at the point 
therefore, the axis of symmetry is at 
.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the axis of symmetry for this particular function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that is equal to the
Step 3: Graph the parabola and plot the vertical line that is equal to the
Step 4: Answer the question.
The vertex is located at the point
What is the end behavior of the following function?
What is the end behavior of the following function?
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the end behavior of the function. This means we will need to examine the graph of the function as the 
values get larger and larger.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the end behavior, 
values as the 
values get larger and larger.
Step 3: Interpret the above graph for the end behavior.
Looking at the graph above there appears to be a vertical asymptote and a horizontal asymptote. The vertical asymptote effects the behavior of the graph as the 
values get close to zero. The horizontal asymptote effects the behavior of the graph as the 
values gets closer to positive infinity and negative infinity. In other words the horizontal asymptote effects the end behavior of the function. As the 
values get larger and larger the function values or end behavior of the function approaches zero.
Step 4: Verify the solution algebraically.
Using the algebraic technique, plug in larger values for 
to see the trend in the function values.
As 
, 
since by definition, one divided by a large number becomes an extremely small number close to zero.
Step 5: Answer question.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the end behavior of the function. This means we will need to examine the graph of the function as the
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the end behavior,
Step 3: Interpret the above graph for the end behavior.
Looking at the graph above there appears to be a vertical asymptote and a horizontal asymptote. The vertical asymptote effects the behavior of the graph as the
Step 4: Verify the solution algebraically.
Using the algebraic technique, plug in larger values for
As
Step 5: Answer question.
What is the periodicity of the following function 
?
What is the periodicity of the following function
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the periodicity of the function
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
II. Algebraically solve using the general formula, 
where
Step 3: Using option II, the algebraic method to solve for the periodicity of the given function.
Therefore the periodicity is
.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the periodicity of the function
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
II. Algebraically solve using the general formula,
Step 3: Using option II, the algebraic method to solve for the periodicity of the given function.
Therefore the periodicity is
Find the interval where 
is decreasing.
Find the interval where
Tap to see back →
This question is testing one's ability to grasp the relationship between the image a function creates graphically and the intervals where the function is increase, decreasing, positive, or negative. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for where the function is decreasing. It is important to understand that when a function is decreasing, the graph exhibits a negative slope.
Step 2: Identify the intervals where the graph is negative (decreasing) and where it is positive (increasing).
Looking at the graph above, it is seen that 
is negative or decreasing roughly between 
. Since the specific function is also included in the question we can use algebra along with the initial assumptions made using the graph to solve the problem.
Step 3: Use algebra to find the decreasing interval of the function.
Looking at the possible answer selections, there are only two possible selections that contain two negative 
values.
Option 1:
Option 2:
Lets convert the fractions into decimals.
Since 
is closer to negative one the decreasing interval of the function is,
.
This question is testing one's ability to grasp the relationship between the image a function creates graphically and the intervals where the function is increase, decreasing, positive, or negative. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for where the function is decreasing. It is important to understand that when a function is decreasing, the graph exhibits a negative slope.
Step 2: Identify the intervals where the graph is negative (decreasing) and where it is positive (increasing).
Looking at the graph above, it is seen that
Step 3: Use algebra to find the decreasing interval of the function.
Looking at the possible answer selections, there are only two possible selections that contain two negative
Option 1:
Option 2:
Lets convert the fractions into decimals.
Since
What is the relative minimum of the function 
.
What is the relative minimum of the function
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the relative minimum of the function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph reaches its valley.
Step 3: For the purpose of this question let's solve using graphing calculator technology.
Graphing the function results in the following graph.
Looking at the above graph the valley occurs roughly at the point 
.
Looking at the possible answer choices, there are only two options that contain a positive 
and 
value.
Option 1:
Option 2:
Since the 
values are the same in these two points lets compare the 
values.
and 
Since
is closer to 
that makes the valley of the function occur at the point, .
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the relative minimum of the function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph reaches its valley.
Step 3: For the purpose of this question let's solve using graphing calculator technology.
Graphing the function results in the following graph.
Looking at the above graph the valley occurs roughly at the point
Looking at the possible answer choices, there are only two options that contain a positive
Option 1:
Option 2:
Since the
Since
Find the end behavior of function 
.
Find the end behavior of function
Tap to see back →
CCSS.MATH.CONTENT.HSF.IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
CCSS.MATH.CONTENT.HSF.IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
Given the following function, find the line of symmetry.
Given the following function, find the line of symmetry.
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the axis of symmetry for this particular function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that is equal to the value of the vertex of the function.
Step 3: Graph the function and plot the vertical line that is equal to the value of the vertex.
Step 4: Answer the question.
The vertex is located at the point 
therefore, the axis of symmetry is at 
.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the axis of symmetry for this particular function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that is equal to the
Step 3: Graph the function and plot the vertical line that is equal to the
Step 4: Answer the question.
The vertex is located at the point
Find the axis of symmetry for the following function.
Find the axis of symmetry for the following function.
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the axis of symmetry for this particular function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that represents a vertical asymptote and the horizontal line that represents a horizontal asymptotes.
Step 3: Graph the function and plot the vertical line that is equal to the value of the vertical line that represents a vertical asymptote and the 
value that is the horizontal line that represents a horizontal asymptotes.
Step 4: Answer the question.
The vertical asymptote occurs at 
and the horizontal asymptote occurs at 
thus making them both axis of symmetry for the given function.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the axis of symmetry for this particular function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that represents a vertical asymptote and the horizontal line that represents a horizontal asymptotes.
Step 3: Graph the function and plot the vertical line that is equal to the
Step 4: Answer the question.
The vertical asymptote occurs at
Given the function 
, what is the 
intercept?
Given the function
Tap to see back →
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the intercept of the function.
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph intersects the -axis.
II. Algebraically, find the intercepts by substituting in zero for and solving for (calculating the -intercept).
Step 3: Choose an approach from Step 2 and perform the necessary actions.
For the purpose of this question let's solve using option I.
Looking at the graph above, the function crosses the -axis at twelve. Therefore, the -intercept is 12.
This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs.
For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify what the question is asking.
This particular question is asking for the
Step 2: Determine the approach to solve the problem.
I. Graphically plot the function using computer technology or graphing calculator.
Then, find where the graph intersects the
II. Algebraically, find the intercepts by substituting in zero for
Step 3: Choose an approach from Step 2 and perform the necessary actions.
For the purpose of this question let's solve using option I.
Looking at the graph above, the function crosses the
Graph the following function.
Graph the following function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify the 
-intercept.
Recall that 
is just a constant,
and any constant raised to zero equals one.
Therefore the 
-intercept occurs at the point 
Step 2: Graph the function using technology and plot the 
-intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify the
Recall that
and any constant raised to zero equals one.
Therefore the
Step 2: Graph the function using technology and plot the
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
Also recall that the natural log and 
cancel each other out which results in,
Step 2: Graph the function using technology and plot the intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
Also recall that the natural log and
Step 2: Graph the function using technology and plot the intercept.
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
Also recall that the natural log and cancel each other out which results in,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
Also recall that the natural log and
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify any intercepts.
Recall that the natural log cannot be less than zero and it has a x-intercept of one.
therefore,
Since the function is,
a table can be created to find the function values.
Step 2: Graph the function using technology and plot the intercept.
Graph the function.
Graph the function.
Tap to see back →
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify the -intercept.
Recall that is just a constant,
and any constant raised to zero equals one.
Therefore the -intercept occurs at the point
Step 2: Graph the function using technology and plot the -intercept.
This question is testing one's ability to graph exponential functions.
For the purpose of Common Core Standards, "graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Identify the
Recall that
and any constant raised to zero equals one.
Therefore the
Step 2: Graph the function using technology and plot the