### All ACT Math Resources

## Example Questions

### Example Question #1 : Graphing

Solve and graph the following inequality:

**Possible Answers:**

**Correct answer:**

To solve the inequality, the first step is to add to both sides:

The second step is to divide both sides by :

To graph the inequality, you draw a straight number line. Fill in the numbers from to infinity. Infinity can be designated by a ray. Be sure to fill in the number , since the equation indicated greater than OR **equal to**.

The graph should look like:

### Example Question #2 : Graphing

Points and lie on a circle. Which of the following could be the equation of that circle?

**Possible Answers:**

**Correct answer:**

If we plug the points and into each equation, we find that these points work only in the equation . This circle has a radius of and is centered at .

### Example Question #3 : Graphing

Which of the following lines is perpendicular to the line ?

**Possible Answers:**

**Correct answer:**

The key here is to look for the line whose slope is the negative reciprocal of the original slope.

In this case, is the negative reciprocal of .

Therefore, the equation of the line which is perpendicular to the original equation is:

### Example Question #1 : Graphing

Let *D* be the region on the (*x*,*y*) coordinate plane that contains the solutions to the following inequalities:

, where is a positive constant

Which of the following expressions, in terms of *, is equivalent to the area of **D*?

**Possible Answers:**

**Correct answer:**

### Example Question #4 : Graphing

A triangle is made up of the following points:

What are the points of the inverse triangle?

**Possible Answers:**

**Correct answer:**

The inverse of a function has all the same points as the original function, except the x values and y values are reversed. The same rule applies to polygons such as triangles.

### Example Question #5 : Graphing

Electrical power can be generated by wind, and the magnitude of power will depend on the wind speed. A wind speed of (in ) will generate a power of . What is the minimum wind speed needed in order to power a device that requires ?

**Possible Answers:**

**Correct answer:**

The simplest way to solve this problem is to plug all of the answer choices into the provided equation, and see which one results in a power of .

Alternatively, one could set up the equation,

and factor, use the quadratic equation, or graph this on a calculator to find the root.

If we were to factor we would look for factors of c that when added together give us the value in b when we are in the form,

.

In our case . So we need factors of that when added together give us .

Thus the following factoring would solve this problem.

Then set each binomial equal to zero and solve for v.

Since we can't have a negative power our answer is .

### Example Question #6 : Graphing

Compared to the graph , the graph has been shifted:

**Possible Answers:**

units up.

units to the right.

units to the left.

units down.

units down.

**Correct answer:**

units to the left.

The inside the argument has the effect of shifting the graph units to the *left*. This can be easily seen by graphing both the original and modified functions on a graphing calculator.

### Example Question #7 : Graphing

The graph of passes through in the standard coordinate plane. What is the value of ?

**Possible Answers:**

**Correct answer:**

To answer this question, we need to correctly identify where to plug in our given values and solve for .

Points on a graph are written in coordinate pairs. These pairs show the value first and the value second. So, for this data:

means that is the value and is the value.

We must now plug in our and values into the original equation and solve. Therefore:

We can now begin to solve for by adding up the right side and dividing the entire equation by .

Therefore, the value of is .

### Example Question #8 : Graphing

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

**Possible Answers:**

**Correct answer:**

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 #9 : Graphing

Which of the following graphs represents the expression ?

**Possible Answers:**

Complex numbers cannot be represented on a coordinate plane.

**Correct answer:**

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 .