Linear Functions - Math
Card 0 of 408
Refer to the line in the above diagram. It we were to continue to draw it so that it intersects the 
-axis, where would its 
-intercept be?
Refer to the line in the above diagram. It we were to continue to draw it so that it intersects the
First, we need to find the slope of the line.
In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown 
.
We can use this to find the 
-intercept 
using the slope formula as follows:
The lower left point has coordinates 
. Therefore, we can set up and solve for 
in this slope formula, setting 
:
First, we need to find the slope of the line.
In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown
We can use this to find the
The lower left point has coordinates
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Determine where the graphs of the following equations will intersect.
Determine where the graphs of the following equations will intersect.
We can solve the system of equations using the substitution method.
Solve for 
in the second equation.
Substitute this value of 
into the first equation.
Now we can solve for 
.
Solve for 
using the first equation with this new value of 
.
The solution is the ordered pair 
.
We can solve the system of equations using the substitution method.
Solve for
Substitute this value of
Now we can solve for
Solve for
The solution is the ordered pair
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Which of the following graphs correctly depicts the graph of the inequality 
Which of the following graphs correctly depicts the graph of the inequality
Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to 
and the y-intercept is equal to 
.
All of the graphs depict a line with slope of 
and y-intercept 
. Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin 
. If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to
All of the graphs depict a line with slope of
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
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Refer to the above diagram. which of the following compound inequality statements has this set of points as its graph?
Refer to the above diagram. which of the following compound inequality statements has this set of points as its graph?
A horizontal line has equation 
for some value of 
; since the line goes through a point with 
-coordinate 3, the line is 
. Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is 
.
A vertical line has equation 
for some value of 
; since the line goes through a point with 
-coordinate 4, the line is 
. Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is 
.
Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is 
.
A horizontal line has equation
A vertical line has equation
Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is
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Which of the following inequalities is graphed above?
Which of the following inequalities is graphed above?
First, we determine the equation of the boundary line. This line includes points 
and 
, so the slope can be calculated as follows:
Since we also know the 
-intercept is 
, we can substitute 
in the slope-intercept form to obtain the equation of the boundary line:
The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either 
or 
. To find out which one, we can test a point in the solution set - for ease, we will choose 
:
0 is less than 3 so the correct symbol is 
.
The inequality is 
.
First, we determine the equation of the boundary line. This line includes points
Since we also know the
The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either
0 is less than 3 so the correct symbol is
The inequality is
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Which equation best matches the graph of the line shown above?
Which equation best matches the graph of the line shown above?
To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.
Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.
Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:
which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:
To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.
Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.
Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:
which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:
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Refer to the above diagram. If the red line passes through the point 
, what is the value of 
?
Refer to the above diagram. If the red line passes through the point
One way to answer this is to first find the equation of the line.
The slope of a line. given two points 
can be calculated using the slope formula
Set 
:
The line has slope 3 and 
-intercept 
, so we can substitute 
in the slope-intercept form:
Now substitute 4 for 
and 
for 
and solve for 
:
One way to answer this is to first find the equation of the line.
The slope of a line. given two points
Set
The line has slope 3 and
Now substitute 4 for
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Refer to the above red line. A line is drawn perpendicular to that line, and with the same 
-intercept. Give the equation of that line in slope-intercept form.
Refer to the above red line. A line is drawn perpendicular to that line, and with the same
First, we need to find the slope of the above line.
The slope of a line. given two points 
can be calculated using the slope formula
Set 
:
The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be 
. Since we want this line to have the same 
-intercept as the first line, which is the point 
, we can substitute 
and 
in the slope-intercept form:
First, we need to find the slope of the above line.
The slope of a line. given two points
Set
The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be
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Line 
includes the points 
and 
. Line 
includes the points 
and 
. Which of the following statements is true of these lines?
Line
We calculate the slopes of the lines using the slope formula.
The slope of line 
is
The slope of line 
is
The lines have the same slope, making them either parallel or identical.
Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form 
, where 
is the 
-coordinate of each point on the line. Therefore, line 
and line 
have equations 
and 
.This makes them parallel lines.
We calculate the slopes of the lines using the slope formula.
The slope of line
The slope of line
The lines have the same slope, making them either parallel or identical.
Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form
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An individual's maximum heart rate can be found by subtracting his or her age from 
. Which graph correctly expresses this relationship between years of age and maximum heart rate?
An individual's maximum heart rate can be found by subtracting his or her age from
In 
form, where y = maximum heart rate and x = age, we can express the relationship as:
We are looking for a graph with a slope of -1 and a y-intercept of 220.
The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.
In
We are looking for a graph with a slope of -1 and a y-intercept of 220.
The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.
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Select the equation of the line perpendicular to the graph of 
.
Select the equation of the line perpendicular to the graph of
Lines are perpendicular when their slopes are the negative recicprocals of each other such as 
. To find the slope of our equation we must change it to slope y-intercept form.
Subtract the x variable from both sides:
Divide by 4 to isolate y:
The negative reciprocal of the above slope: 
. The only equation with this slope is 
.
Lines are perpendicular when their slopes are the negative recicprocals of each other such as
Subtract the x variable from both sides:
Divide by 4 to isolate y:
The negative reciprocal of the above slope:
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Note: Figure NOT drawn to scale.
Refer to the above figure. The circle has its center at the origin; the line is tangent to the circle at the point indicated. What is the equation of the line in slope-intercept form?
Note: Figure NOT drawn to scale.
Refer to the above figure. The circle has its center at the origin; the line is tangent to the circle at the point indicated. What is the equation of the line in slope-intercept form?
A line tangent to a circle at a given point is perpendicular to the radius from the center to that point. That radius, which has endpoints 
, has slope
.
The line, being perpendicular to this radius, will have slope equal to the opposite of the reciprocal of that of the radius. This slope will be 
. Since it includes point 
, we can use the point-slope form of the line to find its equation:
A line tangent to a circle at a given point is perpendicular to the radius from the center to that point. That radius, which has endpoints
The line, being perpendicular to this radius, will have slope equal to the opposite of the reciprocal of that of the radius. This slope will be
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Solve:
Solve:
Use substution to solve this problem:
becomes 
and then is substituted into the second equation. Then solve for 
:
, so 
and 
to give the solution 
.
Use substution to solve this problem:
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Solve for the 
- and 
- intercepts:
Solve for the
To solve for the 
-intercept, set 
to zero and solve for 
:
To solve for the 
-intercept, set 
to zero and solve for 
:
To solve for the
To solve for the
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Write 
in slope-intercept form.
Write
Slope-intercept form is 
.
Slope-intercept form is
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Which of the following is a vertical line?
Which of the following is a vertical line?
A vertical line has infinitely many values of 
but only one value of 
. Thus, vertical lines are of the form 
, where 
is a real number. The only equation of this form is 
.
A vertical line has infinitely many values of
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Which of the following is a horizontal line?
Which of the following is a horizontal line?
A horizontal line has infinitely many values for 
, but only one possible value for 
. Thus, it is always of the form 
, where 
is a constant. Horizontal lines have a slope of 
. The only equation of this form is 
.
A horizontal line has infinitely many values for
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Which of the following is a vertical line?
Which of the following is a vertical line?
A vertical line is one in which the 
values can vary. Namely, there is only one possible value for 
, and 
can be any number. Thus, by this description, the only vertical line listed is 
.
A vertical line is one in which the
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Which of the following has a slope of 0?
Which of the following has a slope of 0?
A line with a slope of zero will be horizontal. A horizontal line has only one possible value for 
, and 
can be any value.
Thus, the only given equation which fits this description is 
.
A line with a slope of zero will be horizontal. A horizontal line has only one possible value for
Thus, the only given equation which fits this description is
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What is the equation of the line displayed above?
What is the equation of the line displayed above?
The equation of a line is 
, with m being the slope of the line, and b being the y-intercept. The y intercept of the line is at 
, so 
.
The x-intercept is at 
, the equation becomes 
, simplification yields 
The equation of a line is
The x-intercept is at
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