Functions and Graphs Worksheet — GCSE — Intermediate

Functions and Graphs worksheet for Key Stage 4 / GCSE (Ages 14-16). Intermediate level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.

Maths•Key Stage 4 / GCSE (Ages 14-16)•Functions and Graphsintermediate

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Exploring Functions and Graphs

Key Stage 4 / GCSE (Ages 14-16) - Intermediate Level

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Instructions: This worksheet focuses on understanding and analysing functions and graphs. You will solve problems related to linear and quadratic functions, interpret graphs, and apply your knowledge to real-world scenarios. Ensure you show all your workings clearly.

Understanding Linear Functions

Solve the following linear equations for x. Show all your workings clearly in the space provided.

1. 3x + 5 = 20

2. 2x - 7 = 13

3. 4x + 9 = 33

4. 5x - 15 = 10

Graphing Linear Functions

For each equation, sketch the graph on the grid provided. Identify the slope and y-intercept. Use the space below each equation to show your calculations.

1. y = 2x + 3

2. y = -x + 4

3. y = 0.5x - 2

4. y = 3x + 1

Quadratic Functions

Solve the following quadratic equations using factorisation. Show all steps clearly.

1. x² - 5x + 6 = 0

2. x² + 3x - 10 = 0

3. x² - 4x - 12 = 0

4. x² + 2x - 8 = 0

Interpreting Graphs

Analyse the graph below and answer the questions. Consider the slope, intercepts, and any turning points.

Point	Coordinate	Description
A	(0	2)
Y-intercept	B	(3
0)	X-intercept	C
(1	4)	Turning point

1. What is the slope of the graph?

2. Describe the shape of the graph.

3. Identify any symmetry in the graph.

Real-World Application

A car travels at a constant speed. The distance (d) it travels is given by the function d = 50t, where t is time in hours. Use this function to solve the problems below.

1. How far does the car travel in 3 hours?

2. If the car travels 200 miles, how long did it travel for?

3. What is the speed of the car?

Function Transformation

Describe the transformations applied to the function f(x) = x² to obtain the following functions. Write your answers in the space provided.

1. g(x) = (x - 3)²

2. h(x) = x² + 4

3. j(x) = -x²

4. k(x) = 2x²

Comparing Functions

Compare the following pairs of functions. Identify key differences in their graphs and properties.

Function 1	Function 2	Difference
y = x²	y = x² + 3	Vertical shift
y = 2x	y = -2x	Reflection
y = x + 1	y = x - 1	Horizontal shift

1. What is the effect of adding a constant to a function?

2. How does multiplying a function by a negative number affect its graph?

Extended Writing: Graph Analysis

Write a detailed analysis of the graph of the function y = x² - 4x + 3. Discuss its vertex, axis of symmetry, intercepts, and overall shape. Use examples to support your analysis.

Analyse the graph of y = x² - 4x + 3. Discuss the vertex, axis of symmetry, intercepts, and overall shape. Use examples to support your analysis. [10 marks]

Answer Key

Activity 1: x = 5 ; x = 10 ; x = 6 ; x = 5

Activity 2: Slope = 2, y-intercept = 3 ; Slope = -1, y-intercept = 4 ; Slope = 0.5, y-intercept = -2 ; Slope = 3, y-intercept = 1

Activity 3: x = 2, 3 ; x = -5, 2 ; x = 6, -2 ; x = -4, 2

Activity 4: Slope = -2 ; Parabola ; Symmetrical about x = 1

Activity 5: 150 miles ; 4 hours ; 50 mph

Activity 6: Shift right 3 units ; Shift up 4 units ; Reflection in x-axis ; Vertical stretch

Activity 7: Vertical shift upwards ; Reflection in x-axis ; Horizontal shift

Activity 8: Vertex at (2, -1), axis of symmetry x = 2, intercepts at (0, 3) and (3, 0), parabolic shape