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Mathematics · 3. Functions and graphs

Function graphs and transformations

Predict parameter effects, then test them using graphs and tables.

35 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

How can a formula change predict the transformation of an entire graph?

Curriculum coverage

Vietnam 2018 and Cambridge: function representations, roots and graph transformations.

Model scope and limitations

A screen graph shows a finite window and cannot alone prove global behaviour.

Learning objectives

  • Connect formulas, tables and graphs.
  • Recognise translations, stretches and reflections.
  • Identify intersections and roots within the viewed domain.

Prerequisite knowledge

  • Cartesian coordinates and substitution
  • Order of operations

Core knowledge

Vertical translation

y = f(x) + k shifts the graph upward by k when k > 0.

Horizontal translation

y = f(x − h) shifts right by h; the sign inside often feels reversed.

Check with a table

Use strategic x-values to verify the graph instead of relying only on its appearance.

Worked example

From y = x² to y = (x−2)²+1: shift right 2 and up 1; the vertex moves from (0,0) to (2,1).

Misconceptions to avoid

  • The sign in f(x−h) produces a right shift by h, opposite to a naïve sign reading.
  • A finite graph window cannot prove the number of roots or global behaviour.

Virtual experiment procedure: Function explorer

  1. Choose a parent function such as f(x) = x² or sin x.
  2. Change one parameter at a time and predict before plotting.
  3. Record at least three key points and describe the transformation.

Safety and cautions

  • The viewing window may hide roots or asymptotes; always check domain and axis scales.

Evidence to collect

  • A table of at least three key points before and after transformation.
  • A transformation description verified by substitution, not appearance alone.
Open the virtual experiment

Diagnostic check

1. How does y = f(x) + 3 compare with y = f(x)?

  1. Shift up 3
  2. Shift right 3
  3. Horizontal compression by 3
Show answer and explanation

Correct answer: Shift up 3

A constant added outside the function changes every y-value.

2. Where do solutions of f(x) = 0 appear on a graph?

  1. x-axis intersections
  2. y-axis intersection
  3. Only maxima
Show answer and explanation

Correct answer: x-axis intersections

On the x-axis, the y-coordinate is zero.

3. How should the effects of parameters a and b be compared?

  1. Change one while holding the other fixed
  2. Change both randomly
  3. Inspect one point only
Show answer and explanation

Correct answer: Change one while holding the other fixed

Changing one variable at a time isolates its effect.

Application practice

1. Where is the vertex of y = (x − 2)² + 1?

  1. (2, 1)
  2. (-2, 1)
  3. (1, 2)
Show answer and explanation

Correct answer: (2, 1)

The form (x − h)² + k has vertex (h, k).

2. What transformation produces y = −f(x)?

  1. Reflection in the x-axis
  2. Reflection in the y-axis
  3. Shift down 1
Show answer and explanation

Correct answer: Reflection in the x-axis

Every y-value changes sign, reflecting the graph in the x-axis.

3. Why check the axis scale when reading a graph?

  1. Equal screen distances may represent different values
  2. Axes always have identical scales
  3. Scale never affects apparent gradient
Show answer and explanation

Correct answer: Equal screen distances may represent different values

Axis scaling can alter the apparent gradient and shape.

Academic references

  1. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

Content publisher
WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer