Whatever values a and b take, the graph of the linear function y = ax + b is a straight line.
The slope a tells how much y changes when x increases by 1, and the larger its size, the steeper the line.
The y-intercept b is the height at which the line meets the y-axis, so changing only b slides the line up or down without changing its tilt.
The blue dashed triangle in the picture starts at the green dot, steps 1 to the right and then rises or falls by a, so its vertical side changes as the slope changes.
A relation that changes at a steady rate
👀 What a linear function is
①A relation in which y changes by the same amount every time x increases by 1 is a linear function, written y = ax + b (a ≠ 0).
②A car travelling at 30 km/h covers 30 km more for every hour that passes, so distance is a linear function of time.
③When a relation with a constant rate of change is plotted point by point, every point lies on one straight line.
Slope and y-intercept
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The graph of y = ax + b and its slope triangle, changing with the slope and the y-intercept
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1
Equation of a linear function
y = ax + b
a is the slope (the change in y when x increases by 1), and b is the y-intercept (the value of y when x = 0).
What the slope tells you
Formula for the slope
a = Δ yΔ x = y2 - y1x2 - x1
Knowing two points on the line, divide the change in y between them by the change in x.
💡 Sign and size of the slope
①If a > 0, the line rises from left to right.
②If a < 0, the line falls from left to right.
③The larger |a|, the steeper the line. Raising the slope from 1 to 3 in the picture makes the vertical side of the dashed triangle three times as long.
④If a = 0, y stays the same as x changes and the line is horizontal; this is a constant function, not a linear one.
⑤Which way the current line tilts is written at the top left of the picture.
Drawing and reading the graph
📝 Steps for drawing the graph
①Mark a point on the y-axis at height b (the green dot in the picture).
②From that point, go 1 to the right and a up (down if a is negative) and mark a second point. The blue dashed triangle shows exactly this move.
③Draw the line through the two points.
④If you need the x-intercept, substitute y = 0 to get x = −b/a.
⑤Two lines with the same slope and different y-intercepts are parallel.
Worked examples
Example 1
Find the slope and the y-intercept of y = 2x − 3.
1
In the form y = ax + b, the coefficient a of x is the slope and the constant b is the y-intercept.
a = 2, b = −3
2
Read the two values directly from the equation.
slope 2, y-intercept −3
▸ slope 2, y-intercept −3
When the equation is already in the form y = ax + b, a and b can be read off without any calculation.
Example 2
Find the slope of the line through (1, 3) and (4, 9).
1
The slope is the increase in y divided by the increase in x.
slope = (9 − 3) / (4 − 1)
2
Work out the numerator and the denominator separately.
= 6 / 3 = 2
▸ 2
Given two points, divide the difference of the y-coordinates by the difference of the x-coordinates. Subtracting in the other order gives the same result.
Summary
Key facts about the graph
y = ax + b → slope a, y-intercept b, x-intercept −b/a
The sign of the slope sets the direction of the line, and its size sets how steep it is.
Grade 8 school test
What is the x-intercept of y = 3x − 6?
①−6
②−2
③2
④3
⑤6
▸ ③ 2
1
The x-intercept is the value of x when y = 0.
0 = 3x − 6
2
Solve this equation for x.
3x = 6 ⇒ x = 2
🎯 Exam points
①Given two points, find the slope first, then substitute one point into y = ax + b to find b.
②Parallel lines have equal slopes and different y-intercepts (a₁ = a₂, b₁ ≠ b₂).
③The x-intercept is where the line meets the x-axis, so substitute y = 0.
④The y-intercept is where the line meets the y-axis, so substitute x = 0.
⑤The signs of the slope and the y-intercept tell you which quadrants the graph passes through.