ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Graph Complex Numbers

Coordinate_pair_1

Point A represents a complex number.  Its position is given by which of the following expressions?

Possible Answers:

Correct answer:

Explanation:

Complex numbers can be represented on the coordinate plane by mapping the real part to the x-axis and the imaginary part to the y-axis.  For example, the expression  can be represented graphically by the point .

Here, we are given the graph and asked to write the corresponding expression.

 not only correctly identifies the x-coordinate with the real part and the y-coordinate with the imaginary part of the complex number, it also includes the necessary 

 correctly identifies the x-coordinate with the real part and the y-coordinate with the imaginary part of the complex number, but fails to include the necessary .

 misidentifies the y-coordinate with the real part and the x-coordinate with the imaginary part of the complex number.

 misidentifies the y-coordinate with the real part and the x-coordinate with the imaginary part of the complex number.  It also fails to include the necessary .

Example Question #2 : How To Graph Complex Numbers

Which of the following graphs represents the expression ?

Possible Answers:

Coordinate_pair_4

Coordinate_pair_3

Coordinate_pair_2

Complex numbers cannot be represented on a coordinate plane.

Coordinate_pair_5

Correct answer:

Coordinate_pair_4

Explanation:

Complex numbers can be represented on the coordinate plane by mapping the real part to the x-axis and the imaginary part to the y-axis.  For example, the expression  can be represented graphically by the point .

Here, we are given the complex number  and asked to graph it.  We will represent the real part, , on the x-axis, and the imaginary part, , on the y-axis.  Note that the coefficient of  is ; this is what we will graph on the y-axis.  The correct coordinates are .

 

Example Question #201 : Algebra

The point  is on the graph of . What is the value of ?

Possible Answers:

Correct answer:

Explanation:

Because points on a graph are written in the form of , and the point given was , this means that  and .

In order to solve for , these values for  and  must be plugged into the given equation. This gives us the following:

We then solve the equation by finding the value of the right side, then dividing the entire equation by 5, as follows:

Therefore, the value of  is .

Example Question #201 : Coordinate Plane

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

 

Possible Answers:

Zero solution

Vertical slope

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 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 #1 : How To Graph A Function

What is the domain of the following function:

 

Possible Answers:

x = all real numbers

x ≠ 2

x ≠ 5

x ≠ –1

x ≠ –2 and x ≠ –3 

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:

5

2

6

3

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 #11 : 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:

3

7

4

5

6

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 #4 : 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:

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