The ACT Math test will always include at least a few problems involving functions. Many of these questions will be in the final one-third of the test where the more difficult problems appear. One of the function topics you’ll want to be familiar with is *function transformations*, which are the explicit rules for how a change in a function’s equation affects its graph. Those changes are summarized in the table below:

Transformation
Change in the Graph of f(x)
f(x + a)
Moves the graph "a" units left
f(x - a)
Moves the graph "a" units right
f(x) + a
Moves the graph "a" unit up
f(x) - a
Moves the graph "a" units down
a·f(x)
Stretches the graph vertically by a factor of "a"
-f(x)
Reflects the graph across the x-axis

Here are some problems involving transformation that are similar to those you could see on the ACT Math test. Give them a try. The solutions appear at the bottom of this post.

1) The graphs of *f*(x) and *g*(x) are shown on the xy-coordinate system below. Which equation relates the functions to each other?

(A) *g*(x) = *f*(x + 3) – 2

(B) *g*(x) = *f*(x – 3) – 2

(C) *g*(x) = *f*(x + 2) – 3

(D) *g*(x) = *f*(x -2) – 3

(E) *g*(x) = *f*(x + 3) + 2

2) If *f*(x) = x^{3}, what is the equation of the function *g*(x) that results when *f*(x) is shifted up six units and to the right five units on the xy-coordinate plane?

(F) *g*(x) = 5x^{3} + 6

(G) *g*(x) = (x + 5)^{3} + 6

(H) *g*(x) = (x – 5)^{3} + 6

(J) *g*(x) = (x + 6)^{3} + 5

(K) *g*(x) = (x – 6)^{3} + 5

3) The graph below is that of a cosine function. Which of the following could be the equation of this function?

(Click on the graph to enlarge)

(A) *f*(x) = cos(2x) + 2

(B) *f*(x) = cos(2x) + 4

(C) *f*(x) = 2cos(x) + 2

(D) *f*(x) = 2cos(x) + 4

(E) *f*(x) = 4cos(x) + 4

4) A circle centered at the origin has a circumference of 8π. What is the equation of the circle that results when this circle is shifted down 3 units and left 4 units on the xy-coordinate plane?

(F) (x – 4)^{2} + (y – 3)^{2} = 4

(G) (x – 4)^{2} + (y – 3)^{2} = 16

(H) (x – 4)^{2} + (y + 3)^{2} = 16

(J) (x + 4)^{2} + (y + 3)^{2} = 16

(K) (x + 4)^{2} + (y – 3)^{2} = 16

5) In the diagram below the function *f*(*x*) is defined to be *f*(*x*) = *x*^{2} – 4*x* + 3. If *g*(*x*) is the function defined by *g*(*x*) = *f*(*x* + *h*) + *k*. What is the value of *h* + *k*?

(A) – 9

(B) – 2

(C) – 1

(D) 1

(E) 6

If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at info@cardinalec.com .

Solutions:

1) (A)

2) (H)

3) (C)

4) (J)

5) (A)

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