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Graphs of functions — KCSE Mathematics

KCSE Mathematics · 106 practice questions · 4 syllabus objectives · 4 revision lessons

37 easy36 medium33 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Plot and sketch graphs of linear, quadratic, cubic and reciprocal functions; identify key features (intercepts, turning point, symmetry)

Use graphs to solve equations graphically and read off function values and intercepts

Identify and sketch transformations of graphs: translations (f(x)+a), stretches (af(x)) and reflections (–f(x), f(–x))

Graphs of functions

Revision Notes

Concise lesson notes for Graphs of functions, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Plotting and Sketching Functions

To plot and sketch graphs of functions, it is essential to identify key features such as intercepts, turning points, and symmetry. Here’s how to approach different types of functions:

Linear Functions:

  • General form: y = mx + c
  • Key features:
    • Intercepts: (0, c) on the y-axis.
    • Symmetry: No symmetry.

Quadratic Functions:

  • General form: y = ax² + bx + c
  • Key features:
    • Turning Point: Vertex of the parabola.
    • Intercepts: Find by setting y = 0 (x-intercepts) and x = 0 (y-intercept).
    • Symmetry: Symmetric about the vertical line through the vertex.

Cubic Functions:

  • General form: y = ax³ + bx² + cx + d
  • Key features:
    • Turning Points: Can have up to two local maxima or minima.
    • Intercepts: Similar to quadratic functions.
    • Symmetry: No symmetry.

Reciprocal Functions:

  • General form: y = 1/x
  • Key features:
    • Intercepts: None (asymptotes at x = 0 and y = 0).
    • Symmetry: Symmetric about the origin.

Understanding these features will help you accurately plot and sketch graphs of these functions.

Key points to remember

  • Identify intercepts for linear and quadratic functions.
  • Locate turning points in quadratic and cubic functions.
  • Recognize symmetry in quadratic and reciprocal functions.
  • Sketch accurate graphs based on key features.
  • Use the general forms of functions for plotting.

Worked example

Plot the quadratic function y = x² - 4.

  • Intercepts: y-intercept at (0, -4); x-intercepts at (-2, 0) and (2, 0).
  • Turning Point: Vertex at (0, -4).
  • Symmetry: Symmetric about the y-axis.

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More lessons in this topic

Lesson 2: Using Graphs to Solve Equations

Objective: Use graphs to solve equations graphically and read off function values and intercepts

Graphs are powerful tools for visualizing equations and finding solutions. To solve an equation graphically, follow these steps:

  1. Plot the function: Start by sketching the graph of the function, such as y = f(x).
  2. Identify intersections: Look for points where the graph intersects the x-axis (roots) and the y-axis (intercepts).
  3. Read values: Use the graph to read off function values and intercepts directly.

For example, to solve the equation y = x^2 - 4 graphically:

  • Plot the function y = x^2 - 4. This is a parabola opening upwards with its vertex at (0, -4).
  • The x-intercepts are found where y = 0. Setting x^2 - 4 = 0 gives x = ±2. Thus, the points (2, 0) and (-2, 0) are the x-intercepts.
  • The y-intercept occurs when x = 0, giving y = -4, so the point (0, -4) is the y-intercept.

In summary, the graph helps us visualize solutions and key values effectively.

  • Plot the function to visualize solutions.
  • Identify x-intercepts where y = 0.
  • Determine the y-intercept by setting x = 0.
  • Read function values directly from the graph.
  • Use intersections to solve equations graphically.

Solve the equation y = 2x + 3 graphically. The y-intercept is (0, 3). The x-intercept is (-1.5, 0).

Lesson 3: Transformations of Graphs

Objective: Identify and sketch transformations of graphs: translations (f(x)+a), stretches (af(x)) and reflections (–f(x), f(–x))

In mathematics, understanding transformations of graphs is essential for analyzing functions. Transformations include translations, stretches, and reflections.

  1. Translations: When a graph is translated, it shifts without changing its shape.

    • Vertical translation: Given by the equation f(x) + a, where 'a' is positive for upward shift and negative for downward shift.
    • Example: If f(x) = x², then f(x) + 3 translates the graph 3 units up.

  2. Stretches: This transformation alters the graph's shape.

    • Vertical stretch: Represented by af(x), where 'a' > 1 stretches the graph vertically, while 0 < a < 1 compresses it.
    • Example: For f(x) = x², 2f(x) = 2x² stretches the graph vertically by a factor of 2.

  3. Reflections: These flip the graph across an axis.

    • Reflection in the x-axis: Represented by -f(x).
    • Reflection in the y-axis: Represented by f(-x).
    • Example: If f(x) = x², then -f(x) = -x² reflects the graph across the x-axis.
  • Translations shift graphs without changing shape.
  • Vertical translations use f(x) + a for upward/downward shifts.
  • Stretches change graph shape with af(x) for vertical adjustments.
  • Reflections flip graphs across axes using -f(x) or f(-x).
  • Understand each transformation to sketch accurately.

Identify the transformation of the graph of f(x) = x² to g(x) = -2f(x + 1).

  • g(x) reflects f(x) across the x-axis.
  • g(x) stretches vertically by a factor of 2.
  • g(x) translates left by 1 unit.
Lesson 4: Understanding Graphs of Functions

Objective: Graphs of functions

Graphs of functions visually represent the relationship between variables. To analyze a function's graph, consider the following key aspects:

  • Axes: The horizontal axis (x-axis) represents the independent variable, while the vertical axis (y-axis) represents the dependent variable.
  • Intercepts: The points where the graph crosses the axes are called intercepts. The x-intercept occurs when y = 0, and the y-intercept occurs when x = 0.
  • Slope: The slope indicates the rate of change of the function. A positive slope rises from left to right, while a negative slope falls.
  • Shape: The shape of the graph can indicate the type of function (linear, quadratic, etc.). For example, a straight line indicates a linear function, while a parabola indicates a quadratic function.

To sketch a basic graph:

  1. Identify the function, e.g., y = 2x + 1.
  2. Determine the intercepts: Set x = 0 to find the y-intercept (1) and set y = 0 to find the x-intercept (-0.5).
  3. Plot these points and draw the line through them.

Understanding these elements helps in interpreting and analyzing functions effectively.

  • Graphs represent relationships between variables visually.
  • Identify x and y intercepts for accurate plotting.
  • Slope indicates the function's rate of change.
  • Different shapes indicate different types of functions.
  • Use key points to sketch graphs accurately.

Sketch the graph of y = x^2 - 4.

  • Identify the y-intercept: Set x = 0, y = -4.
  • Identify x-intercepts: Set y = 0, solve x^2 - 4 = 0, x = ±2.
  • Plot (0, -4), (2, 0), and (-2, 0) and draw a parabola.

Sample Questions

Read 3 questions and answers free. Sign up to access all 106 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER3 marks

State the transformations applied to the graph of y = 2x when it changes to y = 2(x - 3) + 4. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Translate the graph 3 units to the right (1 mk)
Translate the graph 4 units upward (1 mk)
No change in the shape of the graph; it remains a straight line (1 mk)
2
easySHORT ANSWER3 marks

Explain how to determine the y-intercept of the function y = -3x + 4 and provide its value. (3 marks)

Answer & marking scheme

Part (a) — 1 mark
The y-intercept is found by setting x = 0 in the function (1 mk)
Part (b) — 2 marks
Substituting x = 0 into the function gives y = 4 (1 mk)
Therefore, the y-intercept is (0, 4) (1 mk)
3
easySHORT ANSWER2 marks

Define the x-intercept of the function y = 2x + 6 and find its value. (2 marks)

Answer & marking scheme

Part (a) — 1 mark
The x-intercept is the value of x when y = 0 (1 mk)
Part (b) — 1 mark
Set y = 0 and solve: 0 = 2x + 6, leading to x = -3 (1 mk)
4

Identify the turning point and the line of symmetry for the quadratic function y = -3x² + 6x - 2. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Graphs of functions" cover?

Graphs of functions covers Plot and sketch graphs of linear, quadratic, cubic and reciprocal functions; identify key features (intercepts, turning point, symmetry); Use graphs to solve equations graphically and read off function values and intercepts; Identify and sketch transformations of graphs: translations (f(x)+a), stretches (af(x)) and reflections (–f(x), f(–x)), and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Graphs of functions?

HighMarks has 106 Graphs of functions practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Graphs of functions for the KCSE exam?

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Why Practise Graphs of functions?

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