Mathematics • Year 8 • Unit 2 • Lesson 10

The y-Intercept

Build fluency in reading the y-intercept from a graph, picking c straight out of y = mx + c, and computing it for x = 0. One worked example, one guided fill-in, then eight independent problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

The y-intercept is where the line crosses the y-axis — the point (0, c) — and in y = mx + c it is simply c.

Problem. Find the y-intercept of y = 2x + 3 and write it as a coordinate.

Step 1 — Match the equation to y = mx + c.

y = 2x + 3 → m = 2, c = 3.

Reason: the constant on its own (no x next to it) is c.

Step 2 — Read off c.

y-intercept = c = 3.

Reason: c is the y-value when x = 0.

Step 3 — Confirm by substituting x = 0.

y = 2(0) + 3 = 3. ✓

Reason: the y-intercept is always at x = 0.

Step 4 — Write as a coordinate.

(0, 3)

Answer: y-intercept = 3, coordinate (0, 3).

Stuck? Revisit lesson § "The y-Intercept and y = mx + c" — c IS the y-intercept.

2. We do — fill in the missing steps

Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks

Problem. Find the y-intercept of y = −3x + 7 and write it as a coordinate.

Step 1 — Match to y = mx + c: m = ______, c = ______.

Step 2 — Read off c: y-intercept = ______.

Step 3 — Substitute x = 0 to check:

y = −3(0) + 7 = ______ ✓

Step 4 — Coordinate: (______, ______).

Stuck? The minus sign belongs with the x-term, not with c. c is the bare constant 7.

3. You do — independent practice

Show your working under each problem. First four are foundation, next two are standard, last two are extension.

Foundation — read c straight off

3.1 What is the y-intercept of y = 5x + 8? Write as a value and as a coordinate. 1 mark

3.2 What is the y-intercept of y = −2x − 5? 1 mark

3.3 What is the y-intercept of y = 4x (i.e. no +c written)? 1 mark

3.4 What is the y-intercept of y = 6 (a horizontal line)? 1 mark

Standard — substitute x = 0

3.5 Find the y-intercept of y = 3x − 12 by substituting x = 0. Write as a coordinate. 2 marks

3.6 A line crosses the y-axis at (0, −4) and has gradient m = 2. Write the equation of the line in the form y = mx + c. 2 marks

Extension — rearrange first

3.7 Find the y-intercept of 2y = 6x + 10. (Hint: divide both sides by 2 first to get y = …) 2 marks

3.8 A line passes through (0, 5) and (3, 11). Find both m (using the gradient formula) and c (the y-intercept), then write y = mx + c. 2 marks

Stuck on 3.8? One of your points has x = 0, so its y-value IS the y-intercept. Then use the formula on both points for m.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (y = −3x + 7)

m = −3, c = 7. y-intercept = 7. Substitution check: y = 7. Coordinate: (0, 7).

3.1 — y = 5x + 8

y-intercept = 8; coordinate (0, 8).

3.2 — y = −2x − 5

y-intercept = −5; coordinate (0, −5).

3.3 — y = 4x

Hidden c = 0, so y-intercept = 0; coordinate (0, 0) (line passes through the origin).

3.4 — y = 6

Horizontal line; y-intercept = 6; coordinate (0, 6).

3.5 — y = 3x − 12 at x = 0

y = 3(0) − 12 = −12. y-intercept = −12; coordinate (0, −12).

3.6 — Build the equation

m = 2 and c = −4, so equation is y = 2x − 4.

3.7 — 2y = 6x + 10

Divide both sides by 2: y = 3x + 5. y-intercept = 5; coordinate (0, 5).

3.8 — Through (0, 5) and (3, 11)

(0, 5) has x = 0, so c = 5. Gradient m = (11 − 5)/(3 − 0) = 6/3 = 2. Equation: y = 2x + 5.