### All ACT Math Resources

## Example Questions

### Example Question #456 : Advanced Geometry

If this is a sine graph, what is the phase displacement?

**Possible Answers:**

(1/2)*π*

0

*π*

2*π*

4*π*

**Correct answer:**

0

The phase displacement is the shift from the center of the graph. Since this is a sine graph and the sin(0) = 0, this is in phase.

### Example Question #1 : How To Find Transformation For An Analytic Geometry Equation

If this is a cosine graph, what is the phase displacement?

**Possible Answers:**

0

2*π*

(1/2)*π*

4*π*

*π*

**Correct answer:**

*π*

The phase displacement is the shift of the graph. Since cos(0) = 1, the phase shift is *π* because the graph is at its high point then.

### Example Question #2 : How To Find Transformation For An Analytic Geometry Equation

A regular pentagon is graphed in the standard (x,y) coordinate plane. Which of the following are the coordinates for the vertex P?

**Possible Answers:**

**Correct answer:**

Regular pentagons have lines of symmetry through each vertex and the center of the opposite side, meaning the y-axis forms a line of symmetry in this instance. Therefore, point P is negative *b* units in the x-direction, and *c* units in the y-direction. It is a reflection of point (b,c) across the y-axis.

### Example Question #2 : How To Find Transformation For An Analytic Geometry Equation

If *g(x)* is a transformation of *f(x)* that moves the graph of *f(x)* four units up and three units left, what is *g(x) *in relation to* f(x)*?

**Possible Answers:**

**Correct answer:**

To solve this question, you must have an understanding of standard transformations. To move a function along the x-axis in the positive direction, you must subtract the value from the operative x-value. For example, to move a function, f(x), five units to the left would be f(x+5).

To shift a function along the y-axis in the positive direction, you must add the value to the overall function. For example, to move a function, f(x), three units up would be f(x)+3.

The question asks us to move the function, f(x), left three units and up four units. f(x+3) will move the function three units to the left and f(x)+4 will move it four units up.

Together, this gives our final answer of f(x+3)+4.

### Example Question #2 : Transformation

What is the period of the function?

**Possible Answers:**

2*π*

*π*

4*π*

1

3*π*

**Correct answer:**

4*π*

The period is the time it takes for the graph to complete one cycle.

In this particular case we have a sine curve that starts at 0 and completes one cycle when it reaches .

Therefore, the period is