UK K12 GCSE/A-Level: Year 8 KS3 Mathematics - Graphs, Straight Lines, y = mx + c

Learning Objectives

By the end of this topic, students will be able to:

• Understand the concept of a straight line graph and its equation in the form y = mx + c.
• Identify the gradient (m) and y-intercept (c) of a straight line graph from its equation.
• Plot straight line graphs from their equations and identify key features such as x-intercepts and y-intercepts.
• Use the equation of a straight line to solve problems involving real-world contexts.

Core Concepts

A straight line graph is a graphical representation of a linear relationship between two variables, x and y. The equation of a straight line is given by the formula y = mx + c, where m is the gradient (or slope) of the line and c is the y-intercept (the point where the line crosses the y-axis).

The gradient (m) of a straight line represents the rate of change of y with respect to x. It can be positive, negative, or zero, and it determines the steepness of the line. A positive gradient indicates that the line slopes upwards from left to right, while a negative gradient indicates that the line slopes downwards from left to right.

The y-intercept (c) of a straight line is the point where the line crosses the y-axis. It represents the value of y when x is equal to zero.

Worked Examples

Example 1: Plotting a straight line graph

The equation of a straight line is y = 2x + 3. Plot the graph and identify the x-intercept and y-intercept.

To plot the graph, we need to find two points on the line. We can do this by substituting different values of x into the equation and finding the corresponding values of y.

For example, if we substitute x = 0 into the equation, we get y = 2(0) + 3 = 3. This means that the point (0, 3) is on the line.

If we substitute x = 1 into the equation, we get y = 2(1) + 3 = 5. This means that the point (1, 5) is also on the line.

We can plot these two points on a coordinate grid and draw a straight line through them to represent the graph of the equation.

The x-intercept of the line is the point where the line crosses the x-axis. To find this point, we need to set y = 0 and solve for x.

0 = 2x + 3 2x = -3 x = -3/2

So, the x-intercept of the line is the point (-3/2, 0).

The y-intercept of the line is the point where the line crosses the y-axis. We already found this point to be (0, 3).

Example 2: Solving a problem using the equation of a straight line

A car travels at a constant speed of 60 miles per hour. If it starts at a point 20 miles from the city center, how long will it take to reach the city center?

Let x be the time in hours and y be the distance in miles. The equation of the straight line representing the car's journey is y = 60x + 20.

We want to find the time it takes for the car to reach the city center, which is 0 miles from the city center. So, we set y = 0 and solve for x.

0 = 60x + 20 60x = -20 x = -20/60 x = -1/3

So, it will take the car 1/3 hour to reach the city center.

Common Misconceptions

• Many students confuse the gradient (m) with the y-intercept (c). The gradient represents the rate of change of y with respect to x, while the y-intercept represents the value of y when x is equal to zero.
• Some students may think that the equation of a straight line is always in the form y = mx + c. However, the equation can also be written in the form y - c = mx or x - c = my.
• Students may also confuse the x-intercept and y-intercept. The x-intercept is the point where the line crosses the x-axis, while the y-intercept is the point where the line crosses the y-axis.

Exam Tips

• Make sure to read the question carefully and understand what is being asked.
• Identify the key features of the straight line graph, such as the gradient and y-intercept.
• Use the equation of the straight line to solve problems involving real-world contexts.
• Check your answers by substituting the values back into the equation.

MCQs with explanations

MCQ 1: [F]

What is the gradient of the straight line with equation y = 2x + 3?

A) 1 B) 2 C) -2 D) 0

Correct answer: B) 2 Why the distractors fail: A) 1 is the gradient of the line y = x + 3, not y = 2x + 3. C) -2 is the negative of the gradient of the line y = 2x + 3. D) 0 is the gradient of the line y = 3, not y = 2x + 3.

MCQ 2: [H]

What is the equation of the straight line with gradient 2 and y-intercept 3?

A) y = 2x + 3 B) y = 2x - 3 C) y = -2x + 3 D) y = -2x - 3

Correct answer: A) y = 2x + 3 Why the distractors fail: B) y = 2x - 3 has a negative y-intercept. C) y = -2x + 3 has a negative gradient. D) y = -2x - 3 has a negative gradient and a negative y-intercept.

MCQ 3: [F]

What is the x-intercept of the straight line with equation y = 2x + 3?

A) (-3/2, 0) B) (3/2, 0) C) (-1/2, 0) D) (1/2, 0)

Correct answer: A) (-3/2, 0) Why the distractors fail: B) (3/2, 0) is the x-intercept of the line y = -2x + 3, not y = 2x + 3. C) (-1/2, 0) is not the x-intercept of the line y = 2x + 3. D) (1/2, 0) is not the x-intercept of the line y = 2x + 3.

MCQ 4: [H]

A straight line has an equation of the form y = mx + c. What is the value of c when the line passes through the point (1, 2)?

A) 1 B) 2 C) 3 D) 4

Correct answer: C) 3 Why the distractors fail: A) 1 is the value of m, not c. B) 2 is the value of y when x is 1, but it is not the value of c. D) 4 is not the value of c.

MCQ 5: [F]

What is the y-intercept of the straight line with equation y = 2x + 3?

A) 0 B) 3 C) -3 D) 1

Correct answer: B) 3 Why the distractors fail: A) 0 is the y-intercept of the line y = 2x, not y = 2x + 3. C) -3 is the negative of the y-intercept of the line y = 2x + 3. D) 1 is not the y-intercept of the line y = 2x + 3.

Short-answer questions

1. The equation of a straight line is y = 2x + 3. Plot the graph and identify the x-intercept and y-intercept.

2. A car travels at a constant speed of 60 miles per hour. If it starts at a point 20 miles from the city center, how long will it take to reach the city center?

3. What is the gradient of the straight line with equation y = 2x + 3?

4. What is the equation of the straight line with gradient 2 and y-intercept 3?

5. What is the x-intercept of the straight line with equation y = 2x + 3?