Key takeaway: The y-intercept is the place where a graph meets the vertical axis, with x equal to 0, and in a story it is the starting value before the input begins.
Grade 88.F.B.4Functions and Graphs
Definition of Y-Intercept
The y-intercept is the spot where a line cuts through the vertical axis. Every point on that axis has x-coordinate 0, so the crossing always looks like \((0, b)\). The number \(b\) is the output when the input is zero.
In a story problem this number is the starting amount: the fee charged before any miles are driven, the balance before any weeks of saving, the height before any time has passed. To find it, put \(x = 0\) into the equation, or use a known point with the steepness to walk backward to the axis. A line has at most one such crossing unless it is the axis itself.
Formula and notation
In the equation \(y = mx + b\), the constant \(b\) gives the height of the crossing:
\[\text{crossing point} = (0, b)\]
Given a point \((p, q)\) and the steepness \(m\), solve \(b = q – m p\).
Key facts
- The first coordinate of this point is always 0.
- In a real situation, it is the value at the very start, before any input.
- A line parallel to the x-axis, such as y = 4, crosses at that height.
- Setting x = 0 in the equation gives the intercept directly.
Y-Intercept example
A line passes through the points \((4, 11)\) and \((6, 17)\). Find its y-intercept.
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Slope: \(m = \dfrac{17 – 11}{6 – 4} = \dfrac{6}{2} = 3\).
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Use the point \((4, 11)\): \(11 = 3(4) + b\), so \(11 = 12 + b\).
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Solve: \(b = 11 – 12 = -1\).
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Check with the other point: \(3(6) – 1 = 17\). Correct.
The line is \(y = 3x – 1\), and its y-intercept is the point \((0, -1)\).
Common mistake
Watch out
Jonah says the y-intercept of \(y = 3x – 1\) is the point \((-1, 0)\).
Why it is wrong
He swapped the coordinates. On the y-axis the x-value is 0, so the point is \((0, -1)\). The point \((-1, 0)\) lies on the x-axis, and \(3(-1) – 1 = -4\), so it is not even on the line.
Related terms and lessons
- Grade 8 · Linear Functions and ModelingLessonWorksheetQuiz
- Algebra 1 · Linear Functions: Slope-Intercept, Point-Slope and Standard FormLessonWorksheetQuiz
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