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.
Drag the sliders for slope and intercept to watch the line tilt and shift.
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
seegongsik.com
See y = ax + b change as you adjust slope and y-intercept
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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
Work It Out
Example 1
Find the slope and y-intercept of the linear function y = 2x − 3.
1
In y = ax + b, a is the slope and b is the y-intercept.
a = 2, b = −3
2
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).
1
Slope = (change in y) / (change in x).
slope = (9 − 3) / (4 − 1)
2
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
1
The x-intercept is the x-value where y = 0.
0 = 3x − 6
2
Solve for x.
3x = 6 ⇒ x = 2
🎯 Exam Points
①Two points → find slope → plug one point into y=ax+b → find b
②Parallel lines: equal slopes
③x-intercept: set y=0
④y-intercept: set x=0
⑤Slope sign tells which quadrants the line crosses