A10: Gradients & Intercepts
Identify and interpret gradients and intercepts of linear functions graphically and algebraically
๐ Key Concepts
Gradient: The steepness of a line. Found by dividing vertical change by horizontal change.
Intercepts:
- y-intercept: Where the line crosses the y-axis (when x = 0)
- x-intercept: Where the line crosses the x-axis (when y = 0)
๐ Finding Gradient from a Graph
Method:
- Choose two points on the line
- Count the vertical change (up or down)
- Count the horizontal change (right)
- Divide: gradient = verticalโhorizontal
Example 1
A line passes through (1, 2) and (4, 8) on a graph. Find its gradient.
Solution:
Vertical change: 8 - 2 = 6 (go up 6 units)
Horizontal change: 4 - 1 = 3 (go right 3 units)
Gradient = 6โ3 = 2
Signs:
- Going UP = positive vertical change
- Going DOWN = negative vertical change
- Going RIGHT = positive horizontal change
๐ Finding y-intercept
y-intercept: The point where the line crosses the y-axis. At this point, x = 0.
Example 2
Find the y-intercept of y = 3x - 5
Solution:
Substitute x = 0: y = 3(0) - 5 = -5
y-intercept is (0, -5)
Or: In y = mx + c form, c = -5, so y-intercept is (0, -5)
Example 3
A line has gradient 2 and passes through the point (3, 8). Find the y-intercept.
Solution:
Use y = mx + c: 8 = 2(3) + c
8 = 6 + c
c = 2
y-intercept is (0, 2)
๐ Finding x-intercept
x-intercept: The point where the line crosses the x-axis. At this point, y = 0.
Example 4
Find the x-intercept of y = 2x - 6
Solution:
Set y = 0: 0 = 2x - 6
2x = 6
x = 3
x-intercept is (3, 0)
๐ Interpreting Gradient in Context
Real-world meaning: The gradient represents the rate of change between the two variables.
Example 5
A graph shows the cost (ยฃC) of hiring a car for x days. The equation is C = 40x + 50.
a) What does the gradient represent?
b) What does the y-intercept represent?
Solution:
a) Gradient = 40. This represents ยฃ40 per day (daily hire rate)
b) y-intercept = 50. This represents a ยฃ50 fixed fee (insurance/admin)
๐ Writing Equation from Graph
Method:
- Find the gradient (m) from the graph
- Find the y-intercept (c) from the graph
- Write the equation: y = mx + c
Example 6
A line on a graph passes through (0, 3) and (2, 9). Write its equation.
Solution:
y-intercept (0, 3), so c = 3
Gradient = 9 - 3โ2 - 0 = 6โ2 = 3
Equation: y = 3x + 3
โ Practice Questions
Q1: Find the gradient of the line passing through (0, 2) and (4, 10).
Q2: Find the y-intercept of y = 5x - 8.
Q3: Find the x-intercept of y = 3x + 9.
Q4: A line has gradient -2 and passes through (1, 5). Find its y-intercept.
Q5: A distance-time graph shows d = 50t + 100. What does the gradient represent?
Q6: Write the equation of a line with gradient 4 and y-intercept (0, -1).
โ Answers
- 2
- (0, -8)
- (-3, 0)
- (0, 7)
- Speed = 50 (units per hour)
- y = 4x - 1
๐ฏ Exam Tips
- y-intercept: substitute x = 0 into the equation
- x-intercept: substitute y = 0 and solve for x
- Gradient = change in y รท change in x
- Positive gradient = uphill; negative = downhill
- In context, gradient often means "rate" (price per unit, speed, etc.)
๐ง Problem-Solving Strategies
Problem-Solving
In real contexts, the gradient represents a rate (e.g., ยฃ/hour, km/h) and the y-intercept represents a starting value (e.g., fixed fee, initial distance). Always interpret both in context when answering word problems.
Multi-Step Problem
A bath is being filled. After 2 minutes, the water depth is 15 cm. After 5 minutes, it is 30 cm. (a) Find the gradient and interpret it. (b) Find the y-intercept and interpret it. (c) How deep is the bath after 10 minutes?
Solution:
(a) m = 30-15โ5-2 = 5 cm/min (water rises at 5 cm per minute)
(b) y = 5x + c โ 15 = 5(2) + c โ c = 5 cm (water already 5 cm deep when timing started)
(c) y = 5(10) + 5 = 55 cm
โ ๏ธ Common Errors
Watch Out!
1. Wrong: The y-intercept of y = 5x - 3 is 3 Correct: The y-intercept is -3 (the sign is included: y-intercept at (0, -3))
2. Wrong: Gradient = horizontal changeโvertical change Correct: Gradient = vertical changeโhorizontal change = ฮyโฮx
3. Wrong: In C = 30t + 50, the gradient means "30 hours per ยฃ" Correct: The gradient 30 means ยฃ30 per hour (units of y per unit of x)
โ๏ธ 6-Mark Exam Question
Extended Answer
6 marks: Two water tanks are being drained. Tank A: d = 120 - 8t. Tank B: d = 100 - 5t. d = depth in cm, t = time in minutes. (a) Which tank starts fuller? (b) Which drains faster? (c) After how many minutes is Tank A empty? (d) Which tank still has water left after 15 minutes?
(a) Tank A starts at 120 cm, Tank B at 100 cm. Tank A is fuller.
(b) Tank A drains at 8 cm/min (gradient = -8). Tank B drains at 5 cm/min. Tank A drains faster.
(c) 120 - 8t = 0 โ t = 15 minutes
(d) After 15 minutes: Tank A = 120 - 8(15) = 0 cm (empty). Tank B = 100 - 5(15) = 25 cm. Tank B still has water.
Mark scheme: (a) 1 mark. (b) 1 mark with reasoning. (c) 2 marks. (d) 2 marks for both tanks checked.
๐ AO3: Reason & Interpret
Reasoning and Interpretation
A monthly phone bill is B = 0.05t + 12, where B = cost (ยฃ), t = number of texts.
(a) Explain what the gradient and y-intercept represent.
(b) If the bill is ยฃ27, how many texts were sent?
(c) A rival network charges B = 0.08t + 8. After how many texts is it cheaper to stay with the original network?
Answers: (a) Gradient 0.05 = 5p per text. y-intercept 12 = ยฃ12 fixed monthly charge. (b) 27 = 0.05t + 12 โ 0.05t = 15 โ t = 300 texts. (c) 0.05t + 12 < 0.08t + 8 โ 4 < 0.03t โ t > 133.3. Original network cheaper after 134+ texts.
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๐ฌ Video Resources
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