Sketching linear equations
Sketch linear equations using gradient and intercepts, and identify horizontal and vertical lines.
Worked examples
Sketching a line using gradient and y-intercept
Straightforward
Problem
Sketch the line by identifying its gradient and y-intercept.
1
Identify the gradient and y-intercept from the equation.
The equation is in the form where and . The gradient is and the y-intercept is .
2
Plot the y-intercept.
The y-intercept is , so plot the point on the y-axis.
3
Use the gradient to find a second point.
Gradient . From , move right unit and up units to reach the point .
4
Draw the line through the two points.
Plot and , then draw a straight line through both points, extending in both directions.
Answer
y-intercept ; using gradient , a second point is .
Sketching a line using x-intercept and y-intercept
Moderate
Problem
Find the x-intercept and y-intercept of and use them to sketch the line.
1
Find the y-intercept by substituting .
. The y-intercept is .
2
Find the x-intercept by substituting .
. The x-intercept is .
3
Draw the line through both intercepts.
Plot and on the axes, then draw a straight line through both points.
Answer
x-intercept ; y-intercept .
Identifying horizontal and vertical lines
Challenging
Problem
A graph contains two lines. Line A is horizontal and passes through . Line B is vertical and passes through . Write the equation of each line and state its gradient.
1
Identify Line A as a horizontal line and write its equation.
A horizontal line has the same -value for every point. Since Line A passes through , every point on it has . The equation is .
2
Find the gradient of Line A.
For any two points on , the rise is always . So gradient .
3
Identify Line B as a vertical line and write its equation.
A vertical line has the same -value for every point. Since Line B passes through , every point on it has . The equation is .
4
State the gradient of Line B.
A vertical line has zero run for any rise, which would require dividing by zero. The gradient of a vertical line is undefined.
Answer
Line A: , gradient . Line B: , gradient is undefined.
Practise
Q1·Straightforward
What is the gradient of the line ?
Explanation
The equation is in the form where . The gradient is .
Q2·Straightforward
What is the y-intercept of the line ?
Explanation
The equation is in the form where . The y-intercept is .
Q3·Straightforward
For the line , find the value of when .
Explanation
. The point lies on the line.
Q4·Straightforward
State the coordinates of the y-intercept of the line .
Explanation
When : . The y-intercept is .
Q5·Moderate
Find the y-intercept of the line .
Explanation
Substituting : , so . The y-intercept is .
Q6·Moderate
Find the x-intercept of the line .
Explanation
Substituting : , so . The x-intercept is .
Q7·Moderate
Find the x-intercept of the line .
Explanation
Substituting : , so and . The x-intercept is .
Q8·Moderate
Find the y-intercept of the line .
Explanation
Substituting : , so and . The y-intercept is .
Q9·Challenging
What is the gradient of the line ?
Explanation
The line is horizontal. For any two points on it, the rise is , so gradient .
Q10·Challenging
Which equation represents a vertical line passing through ?
Explanation
Every point on a vertical line through has . The equation is . Note: would be a horizontal line, not vertical.
Q11·Challenging
Which equation represents a line with gradient and y-intercept ?
Explanation
With gradient and y-intercept , the equation is . Check: when , ✓
Q12·Challenging
A line has x-intercept and y-intercept . Which equation represents this line?
Explanation
The intercepts give points and . Gradient . With y-intercept , the equation is . Check: when , ✓