**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.

- Rise – UP from Left to Right

- 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.

- The steeper the line, the greater the slope
- 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.

- The less steep the line, the smaller the slope

- Steep slopes have larger positive or negative values

- 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

- Rise: vertical (up and down, like an elevator)

- 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
- is the same slope as 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:
- Label Point 2:
- Substitute the values into the formula, reduce the fraction, write your slope in the form of Rise over Run

- Choose either point to be Point 1 and call the other point, Point 2

- Each point has an x-value and a y-value

- Example: Points (2, -4) and (9,10)
- Point 1: (2, -4)
- and

- Point 2: (9, 10)
- and

- Point 1: (2, -4)

Answer: the slope is , which can be reduced to , 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.

- Slope is the coefficient of the x-value

- 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).
- List your facts:
- x = time in seconds (independent variable, input)
- y = distance in miles (dependent variable, output)
- So, x = 1 (second) and y = 5 (miles), you have coordinate point (1,5)
- Since you started at (0,0), you now have 2 points! Use these 2 points to calculate slope!
- , , ,
- or just “5”

- Answer: The slope of this graph is 5.