grade 8 Linear Functions & Graphs

Linear Functions & Graphs

The graph of y = ax + b is always a straight line. The slope a says how much y changes when x goes up by one, so a bigger slope makes the line steeper. The y-intercept b is where the line meets the y-axis, so changing b slides the line up or down. Change the slope and the intercept and the line tilts and shifts.

Intuition — Constant Change

👀 What is a Linear Function?

①Whenever x increases by 1, y always changes by the same amount

②30 km/h → each extra hour adds 30 km

③Plot this "constant rate" and you get a straight line

Slope and y-intercept

Slope a2

y-intercept b1

Linear Function

y = ax + b

a: slope (rate of change), b: y-intercept (y when x = 0)

Meaning of Slope

Slope Formula

a = Δ yΔ x = y₂ - y₁x₂ - x₁

Given two points, you can compute the slope

💡 What Slope Tells You

①a > 0: line rises to the right

②a < 0: line falls to the right

③Larger |a|: steeper

④a = 0: horizontal line — y does not change

Drawing and Reading the Graph

📝 How to Draw

①Plot y-intercept b on the y-axis

②Use slope a — go right 1, up (or down) by a

③Connect the two points

④If needed, find the x-intercept: set y = 0 → x = -b/a

⑤Same slope and different y-intercepts means the lines are parallel

Work It Out

Example 1

Find the slope and y-intercept of the linear function y = 2x − 3.

In y = ax + b, a is the slope and b is the y-intercept.

a = 2, b = −3

Read off the values.

slope 2, y-intercept −3

▸ slope 2, y-intercept −3

Read a and b directly from the form y = ax + b.

Example 2

Find the slope of the line through (1, 3) and (4, 9).

Slope = (change in y) / (change in x).

slope = (9 − 3) / (4 − 1)

Compute.

= 6 / 3 = 2

▸ 2

Given two points, the slope is the ratio of coordinate differences.

Exam Wrap-up

Core Linear Graph

y = ax + b → slope a, y-intercept b, x-intercept -b/a

Sign and size of slope set the direction and steepness

Grade-8 school exam type

What is the x-intercept of the linear function y = 3x − 6?

①−6

②−2

③2

④3

⑤6

▸ ③ 2

The x-intercept is the x-value where y = 0.

0 = 3x − 6

Solve for x.

3x = 6 ⇒ x = 2

🎯 Exam Points

①Two points → find slope → plug one point into y=ax+b → find b

②Parallel lines: same slope and different y-intercept (a₁ = a₂, b₁ ≠ b₂)

③x-intercept: set y=0

④y-intercept: set x=0

⑤Slope sign tells which quadrants the line crosses

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System of Linear Equations

Linear Functions & Equations

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