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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.



  • 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.

Alg1B 4.2 Slope Pos Neg

Alg1B 4.2 Slope Zero Undef


  • 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.

Alg1B 4.2 Slope Lrg Sm


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

Alg1B 4.2 Slope 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

Alg1B 4.2 Slope Formula


  • 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

Alg1B 4.2 Slope Example

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).
      • 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!
          • x_1=0y_1=0x_2=1y_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”
    • Answer: The slope of this graph is 5.


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