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Study Guide11th GradeLevel 2 · Standard

Free 11th Grade Transformations of Functions

A senior study guide to translating, stretching and reflecting graphs using the form y = a f(b(x − h)) + k. Worked examples track vertices, endpoints, asymptotes and individual points through each transformation, including the order of operations when several are combined, with a six-question self-check and answers.

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Grade
11th Grade
Subject
Math
Topic
Functions
Difficulty
Level 2 · Standard
Estimated time
25 minutes
Curriculum
Texas grade levels (TEKS)
Answers
Not applicable
Format
PDF (A4) + print

Students will practice

  • describing vertical and horizontal translations of a graph
  • describing dilations (stretches) and reflections in the axes
  • finding the image of a point under a sequence of transformations
  • writing the equation of a transformed function and stating its key features

Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Transformations of Functions

  1. 1Polynomials: Division, Factor and Remainder Theorems
  2. 211th Grade Function Notation, Composites and Inverses Worksheet — Level 1
  3. 311th Grade Function Notation, Composites and Inverses Worksheet — Level 2
  4. 411th Grade Function Notation, Composites and Inverses Worksheet — Level 3
Study guide

How to use this study guide

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practice with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

11th Grade students (typically ages 16–17) working on functions. It is pitched at level 2 · standard.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 25 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try Polynomials: Division, Factor and Remainder Theorems.

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Success TutoringWorksheets11th Grade · Mathematics · Functions

Transformations of Functions

What you need to know

Starting from a base graph y = f(x) — such as y = x2, y = √x, y = or y = |x| — every transformed graph can be written in the form y = a f(b(x − h)) + k. Each letter does one job: k shifts the graph up or down, h shifts it left or right, a stretches it vertically (and reflects it in the x-axis if negative), and b squashes it horizontally (and reflects it in the y-axis if negative).

Transformed equationEffect on the graph of y = f(x)Effect on a point (x, y)
y = f(x) + ktranslate up k units (down if k < 0)(x, y + k)
y = f(x − h)translate right h units (left if h < 0)(x + h, y)
y = a f(x)vertical dilation, factor a (stretch if a > 1, squash if 0 < a < 1)(x, ay)
y = −f(x)reflect in the x-axis(x, −y)
y = f(−x)reflect in the y-axis(−x, y)
y = f(bx)horizontal dilation, factor (, y)

Describe y = (x − 3)2 + 2 as a transformation of y = x2

  1. h = 3: translate 3 units to the right.
  2. k = 2: translate 2 units up.
  3. The vertex moves from (0, 0) to (3, 2). The shape is unchanged.

Describe y = −2(x + 1)2 + 5 as a transformation of y = x2

  1. x + 1 = x − (−1), so h = −1: translate 1 unit left.
  2. a = −2: vertical dilation by a factor of 2, then reflection in the x-axis (the parabola now opens downward).
  3. k = 5: translate 5 units up.
  4. The vertex is at (−1, 5) and it is a maximum turning point.

State the domain and range of y = √(x + 4) − 1

  1. The base graph y = √x starts at the endpoint (0, 0) and rises to the right.
  2. x + 4 shifts it 4 units left; − 1 shifts it 1 unit down. New endpoint: (−4, −1).
  3. Domain: x + 4 ≥ 0, so x ≥ −4. Range: y ≥ −1.

The point (2, 5) lies on y = f(x). Find its image on each new graph.

  1. y = f(x − 1) + 3: shift right 1, up 3 → (3, 8)
  2. y = 2f(x): double the y-value → (2, 10)
  3. y = f(2x): halve the x-value → (1, 5)
  4. y = −f(x): change the sign of y → (2, −5)

Find the asymptotes of y = + 1

  1. The base graph y = has asymptotes x = 0 and y = 0.
  2. The 3 on top is a vertical dilation; it does not move the asymptotes.
  3. x − 2 shifts the graph right 2: vertical asymptote x = 2.
  4. + 1 shifts the graph up 1: horizontal asymptote y = 1.

Write the equation of y = x3 after a reflection in the x-axis followed by a translation of 4 units up

  1. Reflect in the x-axis: y = −x3
  2. Translate 4 units up: y = −x[sup 3] + 4
  3. Check with a point: (1, 1) on y = x3 becomes (1, −1) then (1, 3), and −13 + 4 = 3. Correct.
Common mistake: reading the horizontal shift with the wrong sign. y = f(x − 3) moves the graph right 3, and y = f(x + 3) moves it left 3 — the opposite of what the sign seems to say.
Common mistake: misreading a combined horizontal transformation. y = f(2x + 6) is not a shift of 6. Factor first: f(2x + 6) = f(2(x + 3)), which is a horizontal dilation by factor and a shift of 3 units left. In general, deal with dilations and reflections before translations.

Check yourself

  1. y = x2 is transformed to y = (x + 2)2 − 3. State the new vertex.
  2. The point (4, −1) lies on y = f(x). Find its image on y = f(x) + 6.
  3. The point (4, −1) lies on y = f(x). Find its image on y = f(x + 2).
  4. Describe y = 3√x as a transformation of y = √x.
  5. State the asymptotes of y = − 2.
  6. Write the equation of y = x2 after a reflection in the x-axis followed by a translation of 1 unit to the right.

Answers: 1. (−2, −3) 2. (4, 5) 3. (2, −1) 4. a vertical dilation (stretch) by a factor of 3 5. x = −5 and y = −2 6. y = −(x − 1)2

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Created by
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Last reviewed
1 October 2026
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