Function Transformations Worksheet with Answers – Algebra 1 (Free PDF)

Problem 1 Shift Up ★★★

The graph of g(x) = x² + 5 is made from the graph of f(x) = x². Describe the change and name the vertex of g.

Problem 2 Shift Right ★★★

Describe how the graph of g(x) = (x − 4)² differs from the graph of f(x) = x².

Problem 3 Absolute Value, Shifted Down ★★★

Describe how the graph of h(x) = |x| − 3 differs from the graph of f(x) = |x|.

Problem 4 Absolute Value, Shifted Left ★★★

Describe how the graph of f(x) = |x + 2| differs from the graph of |x|.

Problem 5 Evaluate an Absolute Value Function ★★★

Let f(x) = |2x − 7|. Find f(0), f(3), f(5) and f(−1).

Problem 6 Evaluate a Piecewise Function ★★★

Let f(x) = x² − 1 if x < 0, and f(x) = 3x + 2 if x ≥ 0. Find f(−3), f(0) and f(4).

Problem 7 Stretch, Shrink and Flip ★★★

Describe how each graph differs from the graph of f(x) = x².

1. y = 3x²
2. y = 14x²
3. y = −x²

Problem 8 Combine Several Changes ★★★

For g(x) = −2(x − 3)² + 5, list the transformations of f(x) = x², and give the vertex and the direction of opening.

Problem 9 Equation from an Absolute Value Graph ★★★

Use the vertex and one more marked point to find a rule for this V-shaped graph.

Problem 10 Table for f(x − 2) + 1 ★★★

Let f(x) = x² and g(x) = f(x − 2) + 1. Complete a table of g for x = 0, 2, 4.

Problem 11 Read a Piecewise Graph ★★★

Use the graph to find f(−1), f(1), f(2) and f(3). Say which endpoint at x = 2 is open and which is closed.

Problem 12 Absolute Value Equation ★★★

For f(x) = |x − 4| + 1, find every x for which f(x) = 7.

Problem 13 Parking Fee Steps ★★★

A garage charges $4 for the first hour or part of an hour, and $2 for each additional hour or part of an hour, up to 4 hours. Use the graph to find the fee for 1 hour, 2.5 hours and 3.2 hours.

Problem 14 Taxi Fare Rule ★★★

A taxi charges $3.50 plus $2.25 per mile for the first 4 miles, and $1.75 per mile after that. Write C(m) as a piecewise function and find the fare for 2 miles and for 10 miles.

Problem 15 Move a Point ★★★

The point (3, 5) is on the graph of y = f(x). Give the matching point on each graph.

1. y = f(x) + 4
2. y = f(x − 2)
3. y = −f(x)
4. y = 3f(x)

Problem 16 Write the Transformed Parabola ★★★

Write the equation of the graph made from y = x² by moving it 4 units left, reflecting it over the x-axis and moving it 7 units up. Then find the y-value at x = −2.

Problem 17 Absolute Value as Piecewise ★★★

Write f(x) = |x − 3| as a piecewise function without absolute value bars. Check it at x = −1 and x = 6.

Problem 18 Electricity Bill ★★★

An electric company charges $0.12 per kilowatt-hour (kWh) for the first 500 kWh and $0.18 per kWh for each kWh above 500. Write the bill C(k) as a piecewise function and find the bill for 430 kWh and for 760 kWh.

Problem 19 Describe a Transformed Absolute Value ★★★

Describe the transformations that turn f(x) = |x| into g(x) = −3|x + 1| + 2. Name the vertex and find g(−4) and g(2).

Problem 20 Robot on a Track ★★★

A robot drives along a track. Its distance d from a charging dock, in feet, after t seconds follows the V-shaped graph: 6 feet at t = 0, 0 feet at t = 4 and 6 feet at t = 8. Write an equation d = a|t − h| + k and explain what the vertex means.