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.
Printable Worksheets
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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?
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:
where $c$ is a number, positive, negative, or zero, telling us how far up or down the line crosses the vertical axis.
Text description
- A grid with x from −3.5 to 3.5 and y from −2 to 4.5, equal scales, with the origin O marked.
- A plum line sloping gently down to the right crosses the y-axis at the plum point labelled (0, 3).
- A teal line sloping up to the right crosses the y-axis at the teal point labelled (0, 2).
- A gold line at 45 degrees passes through the origin, marked with the point labelled (0, 0).
- A coral line sloping gently down to the right crosses the y-axis below the origin at the coral point labelled (0, −1).
- Every crossing point lies on the y-axis, so its x-coordinate is 0.
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 $(0, c)$.
Text description
- A grid with x from −5 to 7 and y from −3.5 to 7, equal scales; x-axis numbered −4, −2, 4 and y-axis numbered −2, 2, 4, 6.
- A teal straight line slopes up to the right.
- A coral point marks where the line crosses the y-axis, at y = −2, with the callout "crosses here".
- 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 equation of a straight line in gradient-intercept form is:
Text description
- At the top, the equation y = mx + c, with m in teal and c in coral.
- Below, a grid with a teal line sloping up to the right, labelled m = gradient.
- A coral point on the y-axis at (0, 2), where the line crosses, labelled c = y-intercept.
- A dashed plum gradient triangle from (0, 2): a horizontal run of 2 units labelled run, then a vertical rise of 1 unit labelled rise, back up to the line.
By reading the equation you instantly know both the gradient ($m$) and the y-intercept ($c$).
Two methods, choose whichever fits the equation:
- 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 $(0, -3)$.
- Substitute $x = 0$: $2y = 4(0) + 6$.
- Simplify: $2y = 6$.
- Divide both sides by 2: $y = 3$. The y-intercept is $(0, 3)$.
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).
The y-intercept $(0, 50)$ represents the $\$50$ call-out fee, the cost before any hours are worked.
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.
- 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 $y = 2x + 3$.
Text description
- A grid with equal scales and the origin O marked; the x-axis is numbered 1.
- A coral point at (0, 3) on the y-axis, labelled (0, 3).
- A dashed plum gradient triangle from (0, 3): run = 1 to the right, then rise = 2 up, reaching the coral point labelled (1, 5).
- A teal straight line labelled y = 2x + 3 passes through (0, 3) and (1, 5).
Set a 2-minute timer and race through all 10. Click ? to reveal each answer.
?
$c = 5$?
$c = -3$?
$c = 6$?
$c = 0$?
$c = -4$?
$c = 1$?
$c = 0.5$?
$c = 4$?
$c = -3$?
$c = -4$Multiple Choice
Q1. Look at the graph below. What is the y-intercept of the line?
Text description
- A grid with equal scales and the origin O marked; x-axis numbered −4, −2, 2 and y-axis numbered −2, 2, 4, 6.
- A teal straight line slopes down from upper left to lower right.
- A coral point marks where the line crosses the y-axis, at the height 3 (between the 2 and 4 marks).
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.
Text description
- A grid with equal scales and the origin O marked; x-axis numbered −4, −2, 4 and y-axis numbered −4, −2, 2, 4.
- A teal straight line slopes up from lower left to upper right.
- A coral point marks where the line crosses the y-axis, 1 unit below the origin.
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.
(c) Show both methods give the same answer.
Solution
(a) $3y = 6x - 9 \Rightarrow y = 2x - 3$. So $c = -3$.
(b) $x = 0$: $3y = -9 \Rightarrow y = -3$.
(c) Both give y-intercept $= (0, -3)$. ✓
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 $y = 2x + 5$.
- 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.
Lesson Complete!
You've mastered the y-intercept. Next up: putting $m$ and $c$ together to write the equation of a line.