Learn on PengiCalifornia Reveal Math, Algebra 1Unit 3: Linear and Nonlinear Functions

3-2 Rate of Change and Slope

In this Grade 9 California Reveal Math Algebra 1 lesson, students learn to calculate rate of change using the ratio of change in y to change in x, and to find the slope of a line through two points using the slope formula. The lesson covers comparing rates of change across intervals, interpreting positive and negative rates in real-world contexts, and determining whether a function is linear by checking for a constant rate of change. Part of Unit 3: Linear and Nonlinear Functions, this lesson builds the foundational skills students need to analyze and graph linear functions.

Section 1

Slope as rate of change

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

m=ΔyΔx=y2−y1x2−x1,x1≠x2m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}, x_1 \neq x_2

Examples

Find the slope between (−1,4)(-1, 4) and (3,−2)(3, -2): m=−2−43−(−1)=−64=−32m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}.

Section 2

Slope Formula

Property

The slope of the line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. This is the slope formula. The slope is the difference in the y-coordinates divided by the difference in the x-coordinates.

Examples

  • For the points (2,5)(2, 5) and (4,11)(4, 11), the slope is m=11−54−2=62=3m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3.
  • For the points (−3,6)(-3, 6) and (1,4)(1, 4), the slope is m=4−61−(−3)=−24=−12m = \frac{4 - 6}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}.
  • For the points (5,−1)(5, -1) and (−2,3)(-2, 3), the slope is m=3−(−1)−2−5=4−7=−47m = \frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7}.

Explanation

The slope formula is a tool to find a line's steepness without a graph. It calculates the rise by subtracting y-values (y2−y1y_2 - y_1) and the run by subtracting x-values (x2−x1x_2 - x_1), then divides them.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Slope as rate of change

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

m=ΔyΔx=y2−y1x2−x1,x1≠x2m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}, x_1 \neq x_2

Examples

Find the slope between (−1,4)(-1, 4) and (3,−2)(3, -2): m=−2−43−(−1)=−64=−32m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}.

Section 2

Slope Formula

Property

The slope of the line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. This is the slope formula. The slope is the difference in the y-coordinates divided by the difference in the x-coordinates.

Examples

  • For the points (2,5)(2, 5) and (4,11)(4, 11), the slope is m=11−54−2=62=3m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3.
  • For the points (−3,6)(-3, 6) and (1,4)(1, 4), the slope is m=4−61−(−3)=−24=−12m = \frac{4 - 6}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}.
  • For the points (5,−1)(5, -1) and (−2,3)(-2, 3), the slope is m=3−(−1)−2−5=4−7=−47m = \frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7}.

Explanation

The slope formula is a tool to find a line's steepness without a graph. It calculates the rise by subtracting y-values (y2−y1y_2 - y_1) and the run by subtracting x-values (x2−x1x_2 - x_1), then divides them.