Lesson 10 ~25 min Unit 2 · Linear Relationships +90 XP

The y-Intercept

Discover where lines cross the y-axis. Learn to read, calculate and interpret the y-intercept from graphs, equations and real-world contexts.

Think about it: A line crosses the y-axis at $(0, 4)$. What do we know about this point? What is the x-coordinate of any point on the y-axis?

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Think First

warm-up

A line crosses the y-axis at $(0, 4)$. What do we know about this point? What is the x-coordinate of any point on the y-axis?

Record your answer in your workbook.

What is the y-Intercept?

+5 XP

The y-intercept of a straight line is the point where the line crosses (or "intercepts") the y-axis. Because the y-axis is the line $x = 0$, every point on it has an x-coordinate of zero. Therefore, the y-intercept is always written as:

$$\text{y-intercept} = (0,\ c)$$

where $c$ is a number — positive, negative, or zero — telling us how far up or down the line crosses the vertical axis.

Four lines showing positive, negative, and zero y-intercepts.

Always $(0, c)$

The y-intercept always has $x = 0$. Never $(c, 0)$.

Point or number

Write it as the point $(0, c)$ or just the number $c$ — both are correct.

Finding y-Intercept from a Graph

+5 XP

To find the y-intercept from a graph:

Locate where the line crosses the y-axis (the vertical axis).
Read the y-value at that crossing point.
Write the y-intercept as $(0, c)$.

Worked Example 1 — Reading the y-Intercept

Find the y-intercept of the line shown on the graph below.

Locate where the line crosses the y-axis (the vertical axis through the origin).
The line crosses the y-axis 2 units below the origin, so $y = -2$.
Since $x = 0$ at the y-axis, the y-intercept is $(0, -2)$ or simply $c = -2$.

The y-Intercept and $y = mx + c$

+5 XP

The equation of a straight line in gradient-intercept form is:

$$y = mx + c$$

By reading the equation you instantly know both the gradient ($m$) and the y-intercept ($c$).

$y = 3x + 5$

$m = 3$, $c = 5$, y-intercept is $(0, 5)$.

$y = -2x - 1$

$m = -2$, $c = -1$, y-intercept is $(0, -1)$.

Finding y-Intercept from an Equation

+5 XP

Two methods — choose whichever fits the equation:

Method A — Read $c$ directly

If the equation is already in $y = mx + c$ form, the y-intercept is simply the constant $c$.

Method B — Set $x = 0$

Substitute $x = 0$ into any equation and solve for $y$. Works every time.

Worked Example 2 — Using Method A

Find the y-intercept of $y = 4x - 3$.

Identify the form: $y = 4x - 3$ matches $y = mx + c$ with $m = 4$ and $c = -3$.
The y-intercept is $c = -3$, i.e. the point $(0, -3)$.

Worked Example 3 — Using Method B (rearranging needed)

Find the y-intercept of $2y = 4x + 6$.

Substitute $x = 0$: $2y = 4(0) + 6$.
Simplify: $2y = 6$.
Divide both sides by 2: $y = 3$. The y-intercept is $(0, 3)$.

y-Intercept as Starting Value

+5 XP

In real-world problems the y-intercept often represents a starting value, initial amount, or fixed cost — the value of $y$ when $x = 0$ (before anything has happened).

$\text{Cost} = 30h + 50$

Plumber: $\$50$ call-out fee + $\$30$/hr. y-intercept = $50$ (fee before any work).

$\text{Height} = 2t + 10$

Balloon starts at 10 m. y-intercept = $10$ (starting height).

$\text{Value} = -500n + 8000$

Car worth $\$8000$ new. y-intercept = $8000$ (initial value).

The y-intercept $(0, 50)$ represents the $\$50$ call-out fee — the cost before any hours are worked.

Sketching from the y-Intercept

+5 XP

When sketching a line, the y-intercept is your starting point. It gives you the first point to plot. From there, use the gradient to find additional points.

Worked Example 4 — Sketching Using y-Intercept and Gradient

Sketch the line $y = 2x + 3$.

Identify the y-intercept: $c = 3$, so plot $(0, 3)$ on the y-axis.
Identify the gradient: $m = 2 = \frac{2}{1}$, so rise $= 2$, run $= 1$.
From $(0, 3)$, go up 2 and right 1 to reach $(1, 5)$. Plot this point.
Join the points with a straight line and extend both ways. Label $y = 2x + 3$.

Start at $(0, 3)$, then use $m = 2$ (rise 2, run 1) to find the next point.

Anchor first

Always plot the y-intercept first. It anchors your sketch before you use the gradient.

Brain Trainer — Find the y-Intercept

+10 XP

Set a 2-minute timer and race through all 10. Click ? to reveal each answer.

$y = 4x + 5$

$c = 5$

$y = 2x - 3$

$c = -3$

$y = -x + 6$

$c = 6$

$y = 5x$

$c = 0$

$y = -2x - 4$

$c = -4$

$y = \frac{1}{2}x + 1$

$c = 1$

$y = 3x + 0.5$

$c = 0.5$

$2y = 6x + 8$

$c = 4$

$y + 3 = 2x$

$c = -3$

$3y = 9x - 12$

$c = -4$

Multiple Choice

Q1. Look at the graph below. What is the y-intercept of the line?

Q2. What is the y-intercept of $y = 3x + 2$?

Q3. A line crosses the y-axis at $(0, -2)$. What is the value of $c$?

Q4. Which line has a y-intercept of $4$?

Q5. Find the y-intercept of $2y = 4x + 6$. (Hint: substitute $x = 0$.)

Short Answer

SAQ 1. Write down the y-intercept of the line shown below. Give your answer as a coordinate.

Show answer

The line crosses the y-axis 1 unit below the origin. y-intercept = $(0, -1)$, i.e. $c = -1$.

SAQ 2. The equation of a line is $y = -2x + 5$. Find the y-intercept and explain what it means in terms of the graph.

Show answer

$y = -2x + 5$ is in $y = mx + c$ form with $c = 5$. The y-intercept is $(0, 5)$. This means the line crosses the y-axis 5 units above the origin. At the y-intercept, $x = 0$ and $y = 5$.

SAQ 3. A plumber charges a $\$50$ call-out fee plus $\$30$ per hour. The cost equation is $C = 30h + 50$. What is the y-intercept and what does it represent?

Show answer

The y-intercept is $50$ (the constant in $C = 30h + 50$). It represents the $\$50$ call-out fee — the cost before any hours of work are completed ($h = 0$). Even if the plumber does zero hours of work, you still pay $\$50$.

Stretch Challenges

Challenge 1 — Two methods, same answer

A line has equation $3y = 6x - 9$.

(a) Find the y-intercept by rearranging into $y = mx + c$ form.

(b) Find the y-intercept by substituting $x = 0$ directly.

Solution

(a) $3y = 6x - 9 \Rightarrow y = 2x - 3$. So $c = -3$.

(b) $x = 0$: $3y = -9 \Rightarrow y = -3$.

Challenge 2 — Same intercept, different gradients

Two lines both have y-intercept $(0, 2)$. One has gradient $3$, the other has gradient $-1$.

(a) Write the equation of each line.

(b) Explain why lines with the same y-intercept all pass through the same point.

Solution

(a) Line 1: $y = 3x + 2$. Line 2: $y = -x + 2$.

(b) All lines with y-intercept $2$ must pass through $(0, 2)$ because the y-intercept is by definition the point where $x = 0$ and $y = 2$.

Challenge 3 — Find $m$ and $c$ from two points

The line $y = mx + c$ passes through $(0, 5)$ and $(2, 9)$. Find $m$ and $c$.

Solution

From $(0, 5)$: $x = 0$, $y = 5$, so $c = 5$.

Using $(2, 9)$: $9 = m(2) + 5 \Rightarrow 2m = 4 \Rightarrow m = 2$.

So $y = 2x + 5$.

Key Takeaways — The y-Intercept

The y-intercept is where a line crosses the y-axis. At that point, $x = 0$.
It is written as the point $(0, c)$ or just the number $c$.
In $y = mx + c$, the letter $c$ is the y-intercept.
To find from an equation: substitute $x = 0$ and solve for $y$ (or read $c$ directly if already in $y = mx + c$ form).
To find from a graph: read where the line crosses the y-axis.
The y-intercept can be positive, negative, or zero.
In context it often means starting value, initial cost, or beginning amount.

Common pitfall

Don't confuse the y-intercept ($x = 0$) with the x-intercept ($y = 0$). They are different points on different axes!

Lesson Complete!

You've mastered the y-intercept. Next up: putting $m$ and $c$ together to write the equation of a line.

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