Gradient-intercept form
Identify gradient and y-intercept from y = mx + c, interpret them in context, and write equations of linear relationships.
Worked examples
Reading gradient and y-intercept from y = mx + c
Straightforward
Problem
The equation of a line is . State the gradient, the y-intercept, and the coordinates of the point on the line where .
1
Identify the gradient by comparing with .
The coefficient of is , so the gradient is .
2
Identify the y-intercept.
The constant term is , so the y-intercept is . The line crosses the y-axis at .
3
Substitute to find the y-coordinate.
4
State the coordinates of the point.
The point on the line where is .
Answer
Gradient ; y-intercept ; point at is .
Interpreting gradient and y-intercept in a real-world context
Moderate
Problem
A car hire company charges according to , where is the total cost in dollars and is the distance driven in kilometres. Interpret the gradient and y-intercept in this context.
1
Identify the gradient and its units.
The gradient is . Since is in dollars and is in kilometres, the gradient has units of dollars per kilometre.
2
Interpret the gradient in context.
For each additional kilometre driven, the cost increases by . The gradient represents the charge per kilometre: /km.
3
Identify the y-intercept and evaluate when .
The y-intercept is . When : .
4
Interpret the y-intercept in context.
Even when no distance is driven, the cost is . The y-intercept represents the fixed hire fee charged regardless of distance.
Answer
The gradient is the cost per kilometre (/km). The y-intercept is the fixed hire fee of .
Writing an equation from a table of values
Challenging
Problem
A gym offers a membership plan. The table shows the total cost (in dollars) after months:
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Write the equation for in terms of in the form .
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Write the equation for in terms of in the form .
1
Calculate the gradient using two points from the table.
2
Read the y-intercept directly from the table.
When , , so .
3
Write the equation by substituting and .
4
Verify the equation using another point from the table.
When : ✓
Answer
Practise
Q1·Straightforward
The equation of a line is . What is the gradient of this line?
Explanation
Comparing with : the coefficient of is , so the gradient is .
Q2·Straightforward
The equation of a line is . What is the y-intercept of this line?
Explanation
Comparing with : the constant term is , so the y-intercept is .
Q3·Straightforward
The equation of a line is . What is the y-intercept of this line?
Explanation
Comparing with : the constant term is , so the y-intercept is .
Q4·Straightforward
A line has equation . Find the -coordinate of the point on the line where .
Explanation
Substitute :
The point on the line is .
Q5·Moderate
A plumber's charge is modelled by , where is the total cost in dollars and is the number of hours worked. What does the gradient of this equation represent?
Explanation
The gradient is . Since is in dollars and is in hours, increases by for each additional hour worked. The gradient represents the plumber's hourly rate of per hour.
Q6·Moderate
The volume of water (litres) remaining in a tank after minutes of draining is given by . What does the y-intercept of this equation represent?
Explanation
The y-intercept is . Substituting : litres. This is the volume before any draining begins, so the y-intercept represents the initial volume of water in the tank.
Q7·Moderate
A car's distance from a town (in km) is modelled by , where is time in hours. What is the speed of the car in km/h?
Explanation
The equation is . The gradient is . Since is in km and is in hours, the gradient represents the distance covered per hour. The car travels at km/h.
Q8·Moderate
A phone plan's monthly cost is , where is the cost in dollars and is the number of calls made. How much does the plan cost in a month when no calls are made?
Explanation
Substitute :
The monthly cost with no calls is . This is the y-intercept — the fixed monthly fee regardless of usage.
Q9·Challenging
A table of values for a linear relationship is shown:
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What is the gradient of this linear relationship?
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What is the gradient of this linear relationship?
Explanation
Using the points and :
The gradient is . The full equation of this line is .
Q10·Challenging
A table of values for a linear relationship is shown:
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What is the y-intercept of this relationship?
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What is the y-intercept of this relationship?
Explanation
When , , so the y-intercept is . The gradient is , giving the equation . Verify: when , ✓
Q11·Challenging
A cyclist rides home at a steady pace. After minutes she is km from home, and after minutes she is km from home. Assuming the distance varies linearly with time, what is the gradient of the distance-time equation (in km per minute)?
Explanation
Using the points and :
The gradient is km/min. The negative value confirms the cyclist is getting closer to home as time increases.
Q12·Challenging
Water drips into a measuring cylinder at a steady rate. After minute the cylinder contains mL, and after minutes it contains mL. Assuming a linear relationship, how many mL were already in the cylinder before timing began (the y-intercept)?
Explanation
Gradient:
Substitute into :
The y-intercept is mL — there were already mL in the cylinder at . Check: when , ✓
Substitute into :
The y-intercept is mL — there were already mL in the cylinder at . Check: when , ✓