Key takeaway: Slope-intercept form writes a line as y = mx + b, so the steepness and the starting height can be read straight off the equation without any work.
Grade 88.EE.B.6Functions and Graphs
Definition of Slope-Intercept Form
Slope-intercept form is a way of writing a line so that its two most useful facts are in plain sight. The coefficient of \(x\) tells how steep the line is. The constant at the end tells where the line crosses the vertical axis.
That makes graphing quick: put a dot on the axis at the constant, then use the steepness as a rise-over-run instruction to find more dots. The form also suits stories with a starting amount and a steady change, such as a \$24 sign-up fee plus \$7.50 per visit. To use it on an equation written another way, solve for \(y\) first. Teachers like slope-intercept form because comparing two lines takes one glance at two numbers.
Formula and notation
The general pattern is
\[y = mx + b\]
Here \(m\) multiplies \(x\) and gives the steepness, while \((0, b)\) is where the graph meets the vertical axis. In \(y = -\frac{3}{4}x + 6\), the steepness is \(-\frac{3}{4}\) and \(b = 6\).
Key facts
- The number multiplying x gives the steepness, with its sign.
- The number standing alone is the starting height, with its sign.
- Equations like 3x + y = 8 must be solved for y before they match the pattern.
- Two lines with the same m and different b are parallel.
Slope-Intercept Form example
Graph the line \(y = -\frac{3}{4}x + 6\). Name its steepness and starting height, find two more points, and find where it crosses the x-axis.
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Read off \(m = -\frac{3}{4}\) and \(b = 6\), so the line starts at \((0, 6)\).
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Step right 4, down 3: \((4, 3)\). Again: \((8, 0)\).
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At the x-axis \(y = 0\): \(0 = -\frac{3}{4}x + 6\), so \(\frac{3}{4}x = 6\) and \(x = 8\).
The slope is \(-\frac{3}{4}\), the y-intercept is 6, and the line crosses the x-axis at \((8, 0)\).
Common mistake
Watch out
Looking at \(y = -3x + 5\), Kendra says the y-intercept is \(-3\) because it is the number nearest the x.
Why it is wrong
The number attached to \(x\) is the steepness. The starting height is the constant term, here 5, so the line crosses the vertical axis at \((0, 5)\). Test it: replace \(x\) with 0 and the equation gives \(y = 5\). Reading slope-intercept form correctly means separating the two numbers by their jobs.
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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