ACT Math : How to graph a function

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #1 : How To Graph A Function

The Y axis is a _______________ of the function Y = 1/X

 

Possible Answers:

Vertical slope

Zero solution

Asymptote

Equation

Correct answer:

Asymptote

Explanation:

A line is an asymptote in a graph if the graph of the function nears the line as X or Y gets larger in absolute value.  

 

 

 

 

Example Question #1 : How To Graph A Function

Which of the following does NOT represent the graph of a function?

 

 

Possible Answers:

Act_math_159_12

Act_math_159_14

Act_math_159_10

Act_math_159_13

Correct answer:

Act_math_159_13

Explanation:

A function has only one y value for every x value, and must pass the vertical line test. A there exists a vertical line such that it intersects the graph of the equation, then the equation is not a function. The circle is the only answer choice that fails the vertical line test, and so it is not a function.

Example Question #3 : How To Graph A Function

Which of the given functions is depicted below?

 

Act_math_184_01 

 

 

Possible Answers:

Correct answer:

Explanation:

The graph has x-intercepts at x = 0 and x = 8. This indicates that 0 and 8 are roots of the function.

The function must take the form y = x(x - 8) in order for these roots to be true.

The parabola opens downward, indicating a negative leading coefficient. Expand the equation to get our answer.

y = -x(x - 8)

y = -x2 + 8x

y = 8x - x2

Therefore, the answer must be y = 8x - x2

 

 

Example Question #3 : How To Graph A Function

What is the domain of the following function:

 

Possible Answers:

x ≠ –2 and x ≠ –3 

x ≠ 2

x ≠ –1

x = all real numbers

x ≠ 5

Correct answer:

x ≠ –2 and x ≠ –3 

Explanation:

The denominator cannot be zero, otherwise the function is indefinite. Therefore x cannot be –2 or –3.

Example Question #2 : How To Graph A Function

2

The figure above shows the graph of y = f(x). Which of the following is the graph of y = |f(x)|?

Possible Answers:

6

3

2

5

4

Correct answer:

2

Explanation:

One of the properties of taking an absolute value of a function is that the values are all made positive. The values themselves do not change; only their signs do. In this graph, none of the y-values are negative, so none of them would change. Thus the two graphs should be identical.

Example Question #3 : How To Graph A Function

Below is the graph of the function :

 

Which of the following could be the equation for ?

Possible Answers:

Correct answer:

Explanation:

First, because the graph consists of pieces that are straight lines, the function must include an absolute value, whose functions usually have a distinctive "V" shape. Thus, we can eliminate f(x) = x2 – 4x + 3 from our choices. Furthermore, functions with x2 terms are curved parabolas, and do not have straight line segments. This means that f(x) = |x2 – 4x| – 3 is not the correct choice. 

Next, let's examine f(x) = |2x – 6|. Because this function consists of an abolute value by itself, its graph will not have any negative values. An absolute value by itself will only yield non-negative numbers. Therefore, because the graph dips below the x-axis (which means f(x) has negative values), f(x) = |2x – 6| cannot be the correct answer. 

Next, we can analyze f(x) = |x – 1| – 2. Let's allow x to equal 1 and see what value we would obtain from f(1). 

f(1) = | 1 – 1 | – 2 = 0 – 2 = –2

However, the graph above shows that f(1) = –4. As a result, f(x) = |x – 1| – 2 cannot be the correct equation for the function. 

By process of elimination, the answer must be f(x) = |2x – 2| – 4. We can verify this by plugging in several values of x into this equation. For example f(1) = |2 – 2| – 4 = –4, which corresponds to the point (1, –4) on the graph above. Likewise, if we plug 3 or –1 into the equation f(x) = |2x – 2| – 4, we obtain zero, meaning that the graph should cross the x-axis at 3 and –1. According to the graph above, this is exactly what happens. 

The answer is f(x) = |2x – 2| – 4.

Example Question #1 : How To Graph A Function

Which of the following could be a value of f(x) for f(x)=-x^2 + 3?

Possible Answers:

6

3

4

7

5

Correct answer:

3

Explanation:

The graph is a down-opening parabola with a maximum of y=3. Therefore, there are no y values greater than this for this function.

Example Question #7 : How To Graph A Function

Screen_shot_2015-03-06_at_2.14.03_pm

What is the equation for the line pictured above?

Possible Answers:

Correct answer:

Explanation:

A line has the equation

 where  is the  intercept and  is the slope.

The  intercept can be found by noting the point where the line and the y-axis cross, in this case, at  so .

The slope can be found by selecting two points, for example, the y-intercept and the next point over that crosses an even point, for example, .

Now applying the slope formula,

 

 which yields .

Therefore the equation of the line becomes:

Example Question #8 : How To Graph A Function

Which of the following graphs represents the y-intercept of this function?

Possible Answers:

Function_graph_4

Function_graph_2

Function_graph_1

Function_graph_3

Correct answer:

Function_graph_1

Explanation:

Graphically, the y-intercept is the point at which the graph touches the y-axis.  Algebraically, it is the value of  when .

Here, we are given the function .  In order to calculate the y-intercept, set  equal to zero and solve for .

So the y-intercept is at .

Example Question #262 : Coordinate Geometry

Which of the following graphs represents the x-intercept of this function?

Possible Answers:

Function_graph_7

Function_graph_8

Function_graph_5

Function_graph_6

Correct answer:

Function_graph_6

Explanation:

Graphically, the x-intercept is the point at which the graph touches the x-axis.  Algebraically, it is the value of  for which .

Here, we are given the function .  In order to calculate the x-intercept, set  equal to zero and solve for .

So the x-intercept is at .

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