Practice Function Transformation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A function transformation shifts, stretches, compresses, or reflects the graph of a parent function by modifying its formula in a systematic way.

Moving or reshaping a graph without changing its basic shape.

Showing a random 20 of 50 problems.

Example 1

medium
Write g(x)=4x−8+2g(x) = \sqrt{4x - 8} + 2 in the form ab(x−h)+ka\sqrt{b(x - h)} + k and identify each transformation from f(x)=xf(x) = \sqrt{x}.

Example 2

easy
Starting from f(x)=x2f(x)=x^2, what is the vertex of f(x)=(x−4)2f(x)=(x-4)^2?

Example 3

medium
The point (2,5)(2,5) is on y=f(x)y=f(x). Where does it move on y=f(x)+3y=f(x)+3?

Example 4

hard
The graph of y=f(x)y = f(x) has domain [−2,6][-2, 6]. What is the domain of y=f(2x−4)y = f(2x - 4)?

Example 5

hard
Given f(x)=xf(x) = \sqrt{x}, find a single function g(x)g(x) obtained by reflecting ff over the line y=xy = x. State the result.

Example 6

easy
From f(x)=x2f(x) = x^2, where is the vertex of g(x)=(x+5)2−1g(x) = (x+5)^2 - 1?

Example 7

easy
Describe the transformation from f(x)f(x) to f(x−2)f(x-2).

Example 8

easy
The graph of y=f(x)y = f(x) passes through (1,4)(1, 4). Where does the transformed graph y=f(x+2)−3y = f(x+2) - 3 pass through?

Example 9

easy
Describe the transformation from f(x)f(x) to 2f(x)2f(x).

Example 10

medium
From f(x)=xf(x)=\sqrt{x}, describe g(x)=x+4g(x)=\sqrt{x+4}.

Example 11

hard
Write the equation of the function obtained by reflecting f(x)=log⁡2(x)f(x)=\log_2(x) over the yy-axis, compressing horizontally by a factor of 33, and shifting up 55.

Example 12

easy
Describe the transformation from f(x)f(x) to f(−x)f(-x).

Example 13

medium
Starting from f(x)=∣x∣f(x) = |x|, sketch the key features of g(x)=−2∣x+3∣+4g(x) = -2|x + 3| + 4. Give the vertex and direction of opening.

Example 14

challenge
The function g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k takes the point (0,0)(0, 0) on ff to the point (3,−2)(3, -2), and the point (2,4)(2, 4) on ff to (5,−10)(5, -10). If b=1b = 1, find aa, hh, and kk.

Example 15

easy
Describe all transformations applied to f(x)=x2f(x) = x^2 to obtain g(x)=2(x−3)2+1g(x) = 2(x-3)^2 + 1.

Example 16

easy
From f(x)=x2f(x)=x^2, where is the vertex of g(x)=x2+5g(x)=x^2+5?

Example 17

hard
For what value of cc is the graph of y=f(x+c)y = f(x + c) symmetric about the yy-axis, given that y=f(x)y = f(x) is symmetric about the line x=4x = 4?

Example 18

medium
Write the equation after shifting f(x)=x2f(x)=x^2 right 2 and up 3.

Example 19

medium
From f(x)=ln⁡(x)f(x) = \ln(x), write g(x)=ln⁡(x−4)−2g(x) = \ln(x - 4) - 2 and state the vertical asymptote.

Example 20

medium
Compared with y=sin⁡(x)y = \sin(x), what is the period of y=sin⁡(3x)y = \sin(3x)?
← Back to Function Transformation Worked Examples Formula Explained