Describe how y = f(x) becomes y = 1/2f(x + 4) - 2

left 4, vertical compression by 1/2, down 2

Describe how y = f(x) becomes y = 3f(x - 2) + 5

right 2, vertical stretch by 3, up 5

Describe how y = f(x) becomes y = 2f(x + 1) - 2

left 1, vertical stretch by 2, down 2

Describe how y = f(x) becomes y = 3f(x - 2) - 4

right 2, vertical stretch by 3, down 4

Describe how y = f(x) becomes y = 1/2f(x - 3) + 3

right 3, vertical compression by 1/2, up 3

Describe how y = f(x) becomes y = 2f(x - 1) + 3

right 1, vertical stretch by 2, up 3

The graph of f(x) = sqrt(x) is solid and the graph of g is dashed. Describe the transformations from f to g.

reflection in the x-axis, right 2, down 4

Name the parent function and write the equation of the graph.

Parent y = sqrt(x); y = -sqrt(x - 2) - 4

Worksheets · Precalculus

Function transformations worksheet

Replacing x with x minus h moves the graph h to the right, and adding k lifts it. Multiplying it out shows you what that actually does to the equation, which is the part a graph alone will not tell you. A number in front stretches the graph, a minus sign reflects it, and a point (a, b) on it follows every one of those moves.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Transformations of Functions Date Period Transform each function, or answer as asked. Write the shifted function and simplify If $f (x) = - 2 x^{2}$ , write $f (x + 2)$ If $f (x) = x^{2} + 3 x - 3$ , write $f (x + 4) + 1$ If $f (x) = - x^{2} - 2 x$ , write $f (x + 2)$ If $f (x) = x^{2} + 3$ , write $f (x - 5) + 3$ If $f (x) = 2 x^{2} - 3 x + 6$ , write $f (x - 1)$ If $f (x) = x^{2} + 7 x$ , write $f (x - 1) - 4$ If $f (x) = - x^{2} + 3$ , write $f (x + 3)$ Find where a point moves The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = 3 f (x)$ ? The point $(2, 2)$ is on $y = f (x)$ . What point must be on $y = f (x - 2) - 1$ ? The point $(- 2, 6)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 2) - 1$ ? The point $(4, - 3)$ is on $y = f (x)$ . What point must be on $y = - f (x - 4) - 4$ ? The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = - 2 f (x + 2) - 4$ ? The point $(3, - 2)$ is on $y = f (x)$ . What point must be on $y = - f (x + 2) + 2$ ? The point $(- 2, - 1)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 4) - 4$ ? Describe the transformations Describe how $y = f (x)$ becomes $y = f (x) - 5$ Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x + 4) - 2$ Describe how $y = f (x)$ becomes $y = 3 f (x - 2) + 5$ Describe how $y = f (x)$ becomes $y = 2 f (x + 1) - 2$ Describe how $y = f (x)$ becomes $y = 3 f (x - 2) - 4$ Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x - 3) + 3$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Transformations of Functions

Date Period

Transform each function, or answer as asked.

Write the shifted function and simplify
If $f (x) = - 2 x^{2}$ , write $f (x + 2)$
If $f (x) = x^{2} + 3 x - 3$ , write $f (x + 4) + 1$
If $f (x) = - x^{2} - 2 x$ , write $f (x + 2)$
If $f (x) = x^{2} + 3$ , write $f (x - 5) + 3$
If $f (x) = 2 x^{2} - 3 x + 6$ , write $f (x - 1)$
If $f (x) = x^{2} + 7 x$ , write $f (x - 1) - 4$
If $f (x) = - x^{2} + 3$ , write $f (x + 3)$

Find where a point moves
The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = 3 f (x)$ ?
The point $(2, 2)$ is on $y = f (x)$ . What point must be on $y = f (x - 2) - 1$ ?
The point $(- 2, 6)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 2) - 1$ ?
The point $(4, - 3)$ is on $y = f (x)$ . What point must be on $y = - f (x - 4) - 4$ ?
The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = - 2 f (x + 2) - 4$ ?
The point $(3, - 2)$ is on $y = f (x)$ . What point must be on $y = - f (x + 2) + 2$ ?
The point $(- 2, - 1)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 4) - 4$ ?

Describe the transformations
Describe how $y = f (x)$ becomes $y = f (x) - 5$
Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x + 4) - 2$
Describe how $y = f (x)$ becomes $y = 3 f (x - 2) + 5$
Describe how $y = f (x)$ becomes $y = 2 f (x + 1) - 2$
Describe how $y = f (x)$ becomes $y = 3 f (x - 2) - 4$
Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x - 3) + 3$



Sigma Prep · sigmaprep.io/worksheets Name Transformations of FunctionsAnswer keyVersion 1 Date Period Transform each function, or answer as asked. Write the shifted function and simplify If $f (x) = - 2 x^{2}$ , write $f (x + 2)$ $- 2 x^{2} - 8 x - 8$ If $f (x) = x^{2} + 3 x - 3$ , write $f (x + 4) + 1$ $x^{2} + 11 x + 26$ If $f (x) = - x^{2} - 2 x$ , write $f (x + 2)$ $- x^{2} - 6 x - 8$ If $f (x) = x^{2} + 3$ , write $f (x - 5) + 3$ $x^{2} - 10 x + 31$ If $f (x) = 2 x^{2} - 3 x + 6$ , write $f (x - 1)$ $2 x^{2} - 7 x + 11$ If $f (x) = x^{2} + 7 x$ , write $f (x - 1) - 4$ $x^{2} + 5 x - 10$ If $f (x) = - x^{2} + 3$ , write $f (x + 3)$ $- x^{2} - 6 x - 6$ Find where a point moves The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = 3 f (x)$ ? $(- 3, - 6)$ The point $(2, 2)$ is on $y = f (x)$ . What point must be on $y = f (x - 2) - 1$ ? $(4, 1)$ The point $(- 2, 6)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 2) - 1$ ? $(0, 17)$ The point $(4, - 3)$ is on $y = f (x)$ . What point must be on $y = - f (x - 4) - 4$ ? $(8, - 1)$ The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = - 2 f (x + 2) - 4$ ? $(- 5, 0)$ The point $(3, - 2)$ is on $y = f (x)$ . What point must be on $y = - f (x + 2) + 2$ ? $(1, 4)$ The point $(- 2, - 1)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 4) - 4$ ? $(2, - 7)$ Describe the transformations Describe how $y = f (x)$ becomes $y = f (x) - 5$ down 5 Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x + 4) - 2$ left 4, vertical compression by 1/2, down 2 Describe how $y = f (x)$ becomes $y = 3 f (x - 2) + 5$ right 2, vertical stretch by 3, up 5 Describe how $y = f (x)$ becomes $y = 2 f (x + 1) - 2$ left 1, vertical stretch by 2, down 2 Describe how $y = f (x)$ becomes $y = 3 f (x - 2) - 4$ right 2, vertical stretch by 3, down 4 Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x - 3) + 3$ right 3, vertical compression by 1/2, up 3 Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Transformations of FunctionsAnswer keyVersion 1

Date Period

Transform each function, or answer as asked.

Write the shifted function and simplify
If $f (x) = - 2 x^{2}$ , write $f (x + 2)$ $- 2 x^{2} - 8 x - 8$
If $f (x) = x^{2} + 3 x - 3$ , write $f (x + 4) + 1$ $x^{2} + 11 x + 26$
If $f (x) = - x^{2} - 2 x$ , write $f (x + 2)$ $- x^{2} - 6 x - 8$
If $f (x) = x^{2} + 3$ , write $f (x - 5) + 3$ $x^{2} - 10 x + 31$
If $f (x) = 2 x^{2} - 3 x + 6$ , write $f (x - 1)$ $2 x^{2} - 7 x + 11$
If $f (x) = x^{2} + 7 x$ , write $f (x - 1) - 4$ $x^{2} + 5 x - 10$
If $f (x) = - x^{2} + 3$ , write $f (x + 3)$ $- x^{2} - 6 x - 6$

Find where a point moves
The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = 3 f (x)$ ? $(- 3, - 6)$
The point $(2, 2)$ is on $y = f (x)$ . What point must be on $y = f (x - 2) - 1$ ? $(4, 1)$
The point $(- 2, 6)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 2) - 1$ ? $(0, 17)$
The point $(4, - 3)$ is on $y = f (x)$ . What point must be on $y = - f (x - 4) - 4$ ? $(8, - 1)$
The point $(- 3, - 2)$ is on $y = f (x)$ . What point must be on $y = - 2 f (x + 2) - 4$ ? $(- 5, 0)$
The point $(3, - 2)$ is on $y = f (x)$ . What point must be on $y = - f (x + 2) + 2$ ? $(1, 4)$
The point $(- 2, - 1)$ is on $y = f (x)$ . What point must be on $y = 3 f (x - 4) - 4$ ? $(2, - 7)$

Describe the transformations
Describe how $y = f (x)$ becomes $y = f (x) - 5$ down 5
Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x + 4) - 2$ left 4, vertical compression by 1/2, down 2
Describe how $y = f (x)$ becomes $y = 3 f (x - 2) + 5$ right 2, vertical stretch by 3, up 5
Describe how $y = f (x)$ becomes $y = 2 f (x + 1) - 2$ left 1, vertical stretch by 2, down 2
Describe how $y = f (x)$ becomes $y = 3 f (x - 2) - 4$ right 2, vertical stretch by 3, down 4
Describe how $y = f (x)$ becomes $y = \frac{1}{2} f (x - 3) + 3$ right 3, vertical compression by 1/2, up 3

How to do these

If $f (x) = x^{2} + 4 x$ , write $f (x - 3) + 1$

Put x minus 3 in wherever x was.
That gives (x - 3) squared plus 4(x - 3).
Expand and add the 1.

The answer is $x^{2} - 2 x - 2$ .

Where students go wrong

Moving the graph the way the sign looks. f(x - 3) shifts it three to the RIGHT, not left: the parenthesis has to reach zero later, so every x has to be bigger.

Also called shifting functions, translating graphs, f(x - h) + k or horizontal and vertical shifts.

What you can put on this worksheet

Shifting a function and simplifying: If $f (x) = 2 x^{2} + 3 x - 4$ , write $f (x - 2)$ → $2 x^{2} - 5 x - 2$
Where a point goes: The point $(- 4, - 4)$ is on $y = f (x)$ . What point must be on $y = 2 f (x - 1) + 3$ ? → $(- 3, - 5)$
Describe the transformations: Describe how $y = f (x)$ becomes $y = 2 f (x - 1) + 3$ → right 1, vertical stretch by 2, up 3
Describe from two graphs: The graph of $f (x) = \sqrt{x}$ is solid and the graph of $g$ is dashed. Describe the transformations from $f$ to $g$ . → reflection in the $x$ -axis, right 2, down 4
Write g in terms of f from graphs: The graph of $f$ is solid and the graph of $g$ is dashed. Write $g$ in terms of $f$ . → $g (x) = - f (x) + 1$
Name the parent and write the equation: Name the parent function and write the equation of the graph. → Parent $y = \sqrt{x}$ ; $y = - \sqrt{x - 2} - 4$

Questions about these worksheets

Yes. Every function transformations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the function transformations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: shifting a function and simplifying, where a point goes, describe the transformations, describe from two graphs, write g in terms of f from graphs and name the parent and write the equation. Tick as many as you want and set how many of each, or let it spread them evenly.

Moving the graph the way the sign looks. f(x - 3) shifts it three to the RIGHT, not left: the parenthesis has to reach zero later, so every x has to be bigger.

Function transformations worksheet

Select your difficulty

Write the shifted function and simplify

Find where a point moves

Describe the transformations

Write the shifted function and simplify

Find where a point moves

Describe the transformations

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these function transformations worksheets free?+

Do the function transformations worksheets come with an answer key?+

Are these function transformations worksheets printable?+

What grade are these function transformations worksheets for?+

Can I get a different version of the same function transformations worksheet?+

Is this a free alternative to Kuta Software for function transformations?+

Can I choose how hard the function transformations questions are?+

What kinds of question are on a function transformations worksheet?+

What do students get wrong with function transformations?+