1 - Slope

Slope

Let's hit the slopes!



(Take notes as needed on a sheet of notebook paper.)

Slope of Graphs

• Graphs of lines have slope.

• Equations of lines have slope.

• Any two points can determine a line and create a slope.

• We will first discuss the graphs.

What slope does a line have?

• In everyday terms, "slope" is how steep a surface is or how "slanted" it is.

• In Math, slope is how steep or slanted a line is.

• Slope of a line can be positive, negative, 0, or undefined.

• Whether slope is positive or negative requires you to "read" a graph from left to right.

Positive Slopes

• You always read a graph from left to right.

• If you look at the graph on the left side and allow your eyes to follow the line, your eyes will go upward allowing you to see that the line goes up also.

• The graph is said to be increasing.

• The line has a positive slope.

Negative Slopes

• You always read a graph from left to right.

• If you look at the graph on the left side and allow your eyes to follow the line, your eyes will go down allowing you to see that the line goes down also.

• The graph is said to be decreasing.

• The line has a negative slope.

Slope is Zero

• You always read a graph from left to right.

• If you look at the graph on the left side and allow your eyes to follow the line, your eyes will go neither upward or downward.

• The graph is neither increasing or decreasing (neither positive nor negative).

• The line has a slope of ZERO.

Slope is Undefined

• You always read a graph from left to right.

• If you think of the top of the line as being on the left, then the graph goes down.

• However, if you think of the bottom of the line as being on the left, then the graph goes up.

• Since it can not be determined which part of the graph is on the left, we say the slope is undefined.

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What is the slope?

• positive

• negative

• zero

• undefined

What determines "steepness"?

• The lines in the picture all have positive slopes except the green line which has a slope of zero.

• The black line is steeper than the yellow line and is increasing at a faster rate than the yellow line.

What determines "steepness"?

• The lines in the picture all have negative slopes except the green line which has a slope of zero.

• The black line is steeper than the yellow line and is decreasing at a faster rate than the yellow line.

"Steepness" of a line is determined by any two points from the line.

• "m" is used to represent slope

• "rise" is the change in the y-values

• "run" is the change in the x-values

• It does not matter which point is considered to be the "first" or "second" point.

Why is the letter "m" used to represent slope?

Honestly...no one knows.

Some mathematicians have a theory that "m" is used for slope because the French word "monter" means to climb.

Finding Slopes from Graphs

Multiple Choice

What is the  $$\frac{rise}{run}?$$  ?

• $$\frac{6}{5}$$  

• $$\frac{-6}{5}$$  

• $$\frac{5}{6}$$  

• $$\frac{-5}{6}$$  

Multiple Choice

What is the  $$\frac{rise}{run}?$$  ?

• $$\frac{0}{-2\ }\ =\ 0$$  

• $$\frac{0}{2}=\ 0$$  

• $$\frac{2}{0}\$$
  
  slope is undefined

• $$\frac{-2}{0}=\ 0$$  

Multiple Choice

What is the  $$\frac{rise}{run}?$$  ?

• $$\frac{0}{-3\ }=-3$$  

• $$\frac{-3}{0}$$  

  
  
  

   slope is undefined

• $$\frac{3}{0}\$$  

  
  
  

  slope is undefined

• $$\frac{0}{3}=\ 0$$  

Multiple Choice

What is the  $$\frac{rise}{run}?$$  ?

• $$-\frac{5}{3}$$  

• $$\frac{5}{3}$$  
  

• $$\frac{3}{5}\$$

• $$-\frac{3}{5}$$  

Multiple Choice

What is the slope?

• $$m\ =\ \frac{2}{0}\$$  

  slope is undefined

• m =  $$\frac{0}{2}$$  

  slope is undefined

• $$m\ =\ \frac{2}{0}$$
  
  m = 0

• $$m\ =\ \frac{0}{2}$$  

  
  m = 0

Multiple Choice

What is the slope?

• 
  $$m\ =\ \frac{-3}{1}\$$  

  
  

• m =  $$\frac{1}{3}$$  

• $$m\ =\ \frac{3}{1}$$
  

• $$m\ =\ -\frac{1}{3}$$  

Multiple Choice

What is the slope?

• 
  $$m\ =\ -\frac{5}{3}\$$  

  
  

•   $$m\ =\ -\frac{3}{5}$$  

• $$m\ =\ \frac{3}{5}$$
  

• $$m\ =\ \frac{5}{3}$$  

Multiple Choice

What is the slope?

• 
  $$m\ =\ \frac{4}{0}\$$  

  
  
  

  slope is undefined

•   $$m\ =\ \frac{4}{0}=\ 0$$  

• $$m\ =\ \frac{0}{4}\ =\ 0$$
  

• $$m\ =\ \frac{0}{4}$$  

  
  slope is undefined

Finding the Slope of a Line from an Equation

• All equations of lines can be written in Slope-Intercept Form

• "m" represents slope

• "b" is called the "y-intercept" and represents the y-coordinate of the point where the graph of the equation intercepts the y-axis

Multiple Choice

$$y\ =\ \frac{2}{7}x\ +\ 4$$      What is the slope?

• $$\frac{2}{7}$$  

• $$\frac{2}{7}x$$  

• 4

• -4

Multiple Choice

$$y\ =\ -\frac{3}{5}x\ +\ 9$$  


What is the  slope?

• $$-\frac{3}{5}x$$  

• $$-\frac{3}{5}$$  

• 9

• -9

Multiple Choice

$$y\ =\ \frac{2}{7}x\ +\ 4$$      What is the y-intercept?

• $$\frac{2}{7}$$  

• (0, 4)

• 4

• -4

Multiple Choice

$$y\ =\ -\frac{3}{5}x\ +\ 9$$  


What is the  y-intercept?

• (0, 9)

• $$-\frac{3}{5}$$  

• 9

• -9

Multiple Choice

$$y\ =\ \frac{2}{7}x\ +\ 4$$      Where does the line cross the y-axis?

• (0, 4)

• (0, -4)

• (4, 0)

• 4

Multiple Choice

$$y\ =\ -\frac{3}{5}x\ +\ 9$$  


Where does the line cross the y-axis?

• (9, 0)

• (0, 9)

• 9

• (0, -9)

Finding Slope From Two Given Points

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(5,\ 9\right)$$ , what is the rise?

• 3

• -3

• 4

• -4

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(5,\ 9\right)$$ , what is the run?

• 3

• -3

• 4

• -4

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(3,\ 7\right)$$ , what is the rise?

• 2

• -2

• 1

• -1

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(3,\ 7\right)$$ , what is the run?

• 2

• -2

• 1

• -1

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(-6,\ 16\right)$$ , what is the slope ?

• $$-\frac{11}{8}$$  
  

• $$\frac{11}{8}$$

• $$-\frac{8}{11}$$  
  

• $$\frac{8}{11}$$  
  

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(14,\ -3\right)$$ , what is the slope?

• $$-\frac{12}{8}=\ -\frac{3}{2}$$  
  

• $$\frac{12}{8}=\ \frac{3}{2}$$

• $$-\frac{8}{12}=-\frac{2}{3}$$  
  

• $$\frac{8}{12}=\ \frac{2}{3}$$  
  

Multiple Choice

Given two points $$\left(x_1,\ y_1\right)\ =\ \left(2,\ 5\right)$$  and $$\left(x_2,\ y_2\right)\ =\ \left(-1,\ -6\right)$$ , what is the slope?

• $$-\frac{11}{3}$$  
  

• $$\frac{11}{3}$$

• $$\frac{3}{11}$$

• $$-\frac{3}{11}$$