ACT Math: Function Transformations

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?

ACT Transformations 1

 

(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) = x3, 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) = 5x3 + 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?

ACT Transformations 3

(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) = x2 – 4x + 3.  If g(x) is the function defined by g(x) = f(x + h) + k.  What is the value of h + k?

Challenge 1 Number 3

(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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