Y-Intercept
In short: The y-intercept marks where a graph touches the vertical axis, where x is 0, and in a linear model it stands for the value you begin with before anything has changed.
8th GradeFunctions and Graphs8.F.B.4
What is Y-Intercept?
Wherever a line cuts through the vertical axis, that crossing is the y-intercept. Every point on the axis has x equal to 0, so the crossing is always \((0, b)\), and \(b\) tells the output when the input is nothing at all.
Within \(y = mx + b\), the constant \(b\) is that height. In a story it is the amount you begin with: a flat entry fee, an opening balance, a first reading. Changing it lifts or lowers the line while its lean stays put.
How to spot it
- A straight graph is shown and the question wants the height at which it meets the upright axis.
- A story mentions a flat fee or an opening amount paid before the per-unit charge begins.
- An equation reads y = mx + b and the loose number at the end is requested.
- A table has the row with x equal to 0, or lets you run its pattern backward to that row.
The method
- On a graph, read off the height where the line meets the upright axis.
- With an equation, plug in 0 for x and compute y, or just read the constant added at the end.
- In a table, walk backward by subtracting the rate until x reaches 0, or use \(b = y – mx\) with any row.
- Report the result as the point \((0, b)\) and say what it means in the story.
Y-Intercept: solved example
Problem
A skate rental counter saved two receipts: 1 hour cost \(\$5\), and 3 hours cost \(\$9\). Suppose the cost grows steadily with time. Locate the vertical-axis crossing and say what it means.
Solution
Rate: \(\frac{9 – 5}{3 – 1} = 2\) dollars per hour. Use \(b = y – mx = 5 – 2(1) = 3\).
The line is \(y = 2x + 3\) with y-intercept \((0, 3)\): a \(\$3\) fee charged before any time is counted. Check with the second receipt: \(2(3) + 3 = 9\).
Try it
Question 1
What is the y-intercept of \(y = -3x + 14\)?
Show the answer
At \(x = 0\), \(y = 14\), so the point is \((0, 14)\).
Question 2
A line leans with slope \(-5\) and goes through \((4, -2)\). Where does it hit the vertical axis?
Show the answer
\(b = y – mx = -2 – (-5)(4) = 18\), so \((0, 18)\).
Y-Intercept vs. Slope
The y-intercept is a starting position on the vertical axis; the slope is the steady change per step. Two lines can share the same intercept and have different slopes, or share a slope and be parallel with different intercepts. The intercept answers where the line begins, the slope answers how it moves.
Keep going
Lessons
Worksheets and quizzes
- Graphing Linear Equations – Worksheet
- Graphing Linear Equations – Quiz
- Linear Functions as Models – Worksheet
- Linear Functions as Models – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Functions and Graphs