# 4.2 – Slope

Key Terms

• Rate of Change – The measure of how much a dependent variable y changes for a given change in the independent variable x. If the graph of the function is a straight line, the rate of change equals the slope of the line.
• Rise – The vertical distance between two points on a line. It equals the difference between the y-coordinates of the two points.
• Run – The horizontal distance between two points on a line. It equals the difference between the x-coordinates of the two points.
• Slope – A measure of the steepness of a line. The slope equals rise divided by run for any two points on the line. A line that rises from left to right has a positive slope. A line that falls from left to right has a negative slope.

Notes

• Horizontal means “left and right”
• x-axis
• Vertical means “up and down”
• y-axis

• Slopes can rise, fall, be zero (horizontal), or be undefined (vertical)
• Rise – UP from Left to Right
• Positive slope
• Fall – DOWN from Left to Right
• Negative slope
• Straight horizontal lines that do not rise or fall
• Zero slope
• Straight vertical lines that are all rise and no run
• Remember: undefined fractions have a zero in the denominator
• Undefined slopes have a zero run (which in the denominator of the formula: rise over run.

• Slopes can be steep (big) or shallow (small)
• Steep slopes have larger positive or negative values
• The steeper the line, the greater the slope
• Ex: slope = 5 is greater than slope = 2
• Ex: slope = 10 is greater than slope = 0.34
• Ex: slope = -4 is greater than slope = 2
• Even though 4 is negative, the value of 4 is larger (steeper) than 2.
• Shallow slopes have smaller positive or negative values
• The less steep the line, the smaller the slope
• Ex: slope = 2 is less steep than slope = 2.5
• Ex. slope = 5 is less steep than slope = -10
• Even though 10 is negative, the value of 10 is larger (steeper) than 5.

• Parallel Slopes
• Two lines that have the same exact slope are parallel

• Slope Formula
• Rise: vertical (up and down, like an elevator)
• Rise Up: Positive
• Rise Down: Negative
• Run: horizontal (left and right, like walking across a room)
• Run Left: Negative
• Run Right: Positive

• Positive and Negative Rise and Run
• Lines that go UP have a Positive Rise and a Positive Run
• Lines that go DOWN have a Negative Rise and a Positive Run
• Unless you are calculating a line’s run backwards, the Run will always be positive.
• You can do this with careful attention to detail.

• The slope of a line is CONSTANT
• It will never change, no matter what two points you compare!
• The slopes will be the same for ALL points on a line
• $\frac{12}{3}$ is the same slope as $\frac{4}{1}$ if you reduce the fraction!  Both will be a slope of 4 over 1.
• The rise will be 4 and the run will be 1.

• Formula for Finding the Slope Using 2 Points
• Each point has an x-value and a y-value
• Choose either point to be Point 1 and call the other point, Point 2
• It’s usually a good idea to pick the point on the Left as Point 1
• Label Point 1: $(x_1,y_1)$
• Label Point 2: $(x_2,y_2)$
• Substitute the values into the formula, reduce the fraction, write your slope in the form of Rise over Run

• Example: Points (2, -4) and (9,10)
• Point 1: (2, -4)
• $x_1=2$ and $y_1=-4$
• Point 2: (9, 10)
• $x_2=9$ and $y_2=10$

Answer: the slope is $\frac{14}{7}$, which can be reduced to $\frac{7}{1}$, which can be written as just “7”

• Equation of a Line
• Formula: y = mx + b
• Slope: m
• y-intercept (where the line crosses the y-axis): b
• b can be 0 if the line crosses the y-axis at the origin (0, 0)
• Any x-value (point) on the line: x
• Any y-value (point) on the line: y

• Identify Slope in an Equation
• Slope is the coefficient of the x-value
• Ex. y = 3x + 2 has a slope of 3.
• Ex. y = -12x has a slope of -12.
• Ex. y = 4(x – 2) + 6 has a slope of 4.
• In this case, distribute the 4 into the parenthesis where the x-value is to simplify: y = 4x – 8 + 6.  You can also add -8 + 6 to find the y-intercept (b) of -2.

• Real World Example
• A graph is constructed such that time (in seconds) is the x-variable and distance (in miles) is the y-variable.
• If you plot the distance that a space shuttle travels at a speed of 5 miles per second, what is the slope of the graph?
• Note: in the real world, most lines START at the ORIGIN (0,0).
• $x_1=0$$y_1=0$$x_2=1$$y_2=5$
• $Slope=\frac{y_2-y_1}{x_2-x_1}$
• $Slope=\frac{5-0}{1-0}$
• $Slope=\frac{5}{1}$ or just “5”