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Precalculus and Trigonometry · 35 minutes

Suggested grades 11–12 · Unit 1: Functions, transformations and rational graphs · Lesson 1 of 2

Transformations of a Function

Separate vertical changes from changes inside the input.

Mr. S Teaches · Original starter lesson · HS-PRE-01 · Grade range is planning guidance.

Watch, notice, discuss

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After watching

How does each change in the formula move a known point on the parent graph?

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Learning objective

Predict and verify transformed points and key graph features.

Before you begin

  • Function notation
  • Quadratic and absolute-value graphs

Gather your materials

  • Paper and pencil
  • Graph paper
  • Calculator for checking results

Language to know

vertical shift
Adding a constant to every output.
horizontal shift
Changing which input produces a given feature.
reflection
A transformation reversing an orientation across an axis.

Understand the concept

For f(x)=x², g(x)=−2f(x−3)+1 shifts the vertex right 3, scales vertical distances by 2, reflects across the horizontal axis and shifts up 1. Changes inside the input must be interpreted by solving for the old input. A point(u,v) on f becomes(u+3,−2v+1) on g. Order matters when several operations are combined.

Teach it step by step

  1. List f points(−1,1),(0,0),(1,1).
  2. Apply the point rule to find three g points and locate its vertex.
  3. Evaluate the algebraic formula at those inputs to verify the mapping.
  4. Compare f(x)+3 with f(x+3), using their vertices rather than relying on verbal shortcuts.

Make the thinking visible

Worked example or model response

The transformed points are(2,−1),(3,1),(4,−1); vertex(3,1), opening downward. f(x)+3 has vertex(0,3), while f(x+3) has vertex(−3,0).

Guided practice

For f(x)=|x|, map three points under h(x)=f(x+2)−4 and sketch the result.

Independent practice

1. State vertex and opening of y=3(x+1)²−2. 2. Explain how y=−f(x) differs from y=f(−x) for a nonsymmetric function.

Quick assessment

Why does x−3 inside the function move the vertex right rather than left?

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: Vertex(−2,−4); points(−3,−3),(−2,−4),(−1,−3).
  2. Independent 1: Vertex(−1,−2), opening upward with vertical scale 3. 2: The first reflects outputs across the x-axis; the second reflects inputs across the y-axis.
  3. Quick check: The old vertex input 0 occurs when x−3=0, hence x=3.

Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.

Make it work for your learner

More support

Use point-mapping tables before a full graph and change one parameter at a time.

Extension challenge

Choose a function that is neither even nor odd and compare both reflection graphs.

Verified references · Selected contributions

Standards connections

Subject-content standards

These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.

Contributes to Common Core standard

HSF.BF.B.3

Common Core · High school · Functions

Regular high school mathematics standard

Explain and verify effects of shifts, scaling and reflections on function graphs.

Read the official standard ↗

This selected transformation lesson does not cover every parameter form or the full even/odd-function analysis.

Verified 2026-11-04. Selected contribution, not full attainment.