LGeoCh3.3: Slope of lines

Match

Match the following

• 12

• 13

• 14

• 17

Positive Slope

(Increase Line)

Negative Slope

(Decrease Lined

Zero Slope

(Horizontal Line)

Undefined (Vertical Line)

• Positive Slope

  (Increase Line)

• Negative Slope

  (Decrease Lined

• Zero Slope

  (Horizontal Line)

• Undefined (Vertical Line)

Match

Match the following

• 12

• 16

• 14

• 17

Positive Slope

Increase Line

Negative Slope

Decrease Lined

Zero Slope

Horizontal Line

Undefined

Vertical Line

• Positive Slope

  Increase Line

• Negative Slope

  Decrease Lined

• Zero Slope

  Horizontal Line

• Undefined

  Vertical Line

Match

Match the following

• 16

  Decrease Lined

• 14

  Horizontal Line

• 15

  Increase Lines

• 17

  Vertical Line

Negative Slope

Zero Slope

Positive Slope

Undefined

• Negative Slope

• Zero Slope

• Positive Slope

• Undefined

Match

Match the following

• Decrease Lines

• Horizontal Lines

• Increase Lines

• Vertical Lines

Negative Slope

Zero Slope

Positive Slope

Undefined

• Negative Slope

• Zero Slope

• Positive Slope

• Undefined

​What is "Slope"?

​The slope of a line is the change in the y values over the change in the x values. In other words, the slope of a line is how much the line rises or falls over how much the line moves to the right. Usually, we measure this using known points on a line using the formula to the right. In linear equation forms, it is usually represented by m.

Multiple Choice

Who is correct? Terrell or Hale?

• Terrell

• Hale

Multiple Choice

$$m=\frac{y_2-y_1}{x_2-x_1}$$ ; m = slope of the line $$\overline{AB}$$ passed through $$A\left(x_{1,\ }y_1\right)\$$ and $$B\left(x_{2,\ }y_2\right)\$$ is also .

• $$m=\frac{y_1-y_2}{x_1-x_2}$$

• $$m=\frac{y_1-y_2}{x_2-x_1}$$

• $$m=\frac{y_2-y_1}{x_1-x_2}$$

• $$m=\frac{x_2-x_1}{y_2-y_1}$$

• $$m=\frac{y_2-x_2}{y_1-x_1}$$

Multiple Choice

$$m=\frac{y_2-y_1}{x_2-x_1}$$ ; m = slope of the line $$\overline{AB}$$ is same as $$m=\frac{y_1-y_2}{x_1-x_2}$$ . $$A\left(x_{1,\ }y_1\right)\$$ & $$B\left(x_{2,\ }y_2\right)\$$

• $$m=\frac{y_1-y_2}{x_1-x_2}$$

• $$m=\frac{y_1-y_2}{x_2-x_1}$$

• $$m=\frac{y_2-y_1}{x_1-x_2}$$

• $$m=\frac{x_2-x_1}{y_2-y_1}$$

• $$m=\frac{y_2-x_2}{y_1-x_1}$$

Multiple Choice

$$m=\frac{y_2-y_1}{x_2-x_1}$$ ; m = slope of the line $$\overline{AB}$$ is same as $$m=\frac{y_1-y_2}{x_1-x_2}$$ . $$A\left(x_{1,\ }y_1\right)\$$ & $$B\left(x_{2,\ }y_2\right)\$$

• True

• False

• None

Multiple Choice

$$m=\frac{y_2-y_1}{x_2-x_1}$$ ; the slope m is same as $$m=\frac{y_1-y_2}{x_1-x_2}$$ . $$A\left(x_{1,\ }y_1\right)\$$ & $$B\left(x_{2,\ }y_2\right)\$$ because of the fact: $$a-b=-\left(b-a\right)$$ or $$a=-\left(-a\right)$$

• $$y_2-y_1=-\left(y_1-y_2\right)$$

  $$x_2-x_1=-\left(x_1-x_2\right)$$

• $$a-b=b-a$$

• None

Lesson Geo3.3: Slope of Lines

Rise over Run...

​NOTE:

Slopes will usually appear as fractions and must always be simplified!​

​Types of Slopes

​There are four types of slopes:

​

​Negative slopes move down when read from left to right

​

​Positive slopes move up when read from left to right

​

​Undefined slopes are vertical (straight up and down) lines

​

​Zero slopes are horizontal (straight left-to-right) lines

​

​

​Types of Slopes

​​In linear equations...

​

if the value of m is negative, then the line should have a negative slope.

​

​if the value of m is positive, then the line should have a positive slope.

​

if the statement says x = n (where n is any numerical value), then the slope is undefined.

​

if the statement says y = n (where n is any numerical value), then the slope is zero.

​

​

​Practice!

​Given the following points, find whether the lines have a positive slope, negative slope, undefined slope, or zero slope:

​

• ​(9, 2) and (2, -3)

• ​(-1, 0) and (3, 2)

• ​(8, 4) and (-5, -4)

Multiple Choice

What kind of slope does this line have?

• positive

• negative

• undefined

• zero

Multiple Choice

What kind of slope does this line have?

• positive

• negative

• undefined

• zero

Multiple Choice

Find the slope of the line.

• -2

• 2

• -1

• 1

Multiple Choice

Find the slope of the line that passes through

(-4, 7) and (-6, -4)

• 2/11

• -2/11

• 11/2

• -11/2

Multiple Choice

What type of slope is shown by the stairs in the image?

• positive

• negative

• undefined

• zero

Multiple Choice

What type of slope is shown in the image?

• Positive

• Negative

• Undefined

• Zero

Multiple Choice

What type of slope is Mario running in the picture?

• Positive

• Negative

• Undefined

• Zero

Slope Review

Review of slope from equations, zero slope, undefined slope, and parallel/perpendicular slopes.

Linear Equations Review

Slope-Intercept Form

Point-Slope Form

Standard Form

Parallel & Perpendicular Lines

​Parallel and Perpendicular Lines

​Two lines are parallel to each other if they have the same slopes.

​

​Two lines are perpendicular to each other if their slopes are opposite reciprocals. In order for slopes to be opposite reciprocals, they must be opposite signs (one positive and one negative) and flipped fractions of each other (2/3 is the reciprocal of 3/2).

Slope-Intercept: y = mx+b

• Used for graphing linear equations

• Need a slope (m) and y-intercept (b)

Slope from slope-intercept form

If the equation is in slope-intercept form (solved for y), then the slope is the number IN FRONT OF the x. NOT the x.

Example: y = mx + b, m is the slope. NOT mx.

mx + b = y, m is still the slope.

Multiple Choice

Find the equation that corresponds to the graph.

• y = -3/4x + 4

• y = 4/3x + 4

Fill in the Blank

In the equation, y = -4x + 2, what is the slope?

Point-slope Form: y - y₁ = m( x - x₁)

• Useful when you are given a point (x₁, y₁) and slope (m)

• Plug in the opposite values of points into formula

Multiple Choice

What is the slope in the equation y = 7x - 4

• 7

• 7x

• -4

• 1

Multiple Choice

Given m = 3 and (-2 , 5) write the equation in point-slope form.

• y + 5 = 3 (x - 2)

• y - 5 = 3 (x + 2)

Multiple Select

Choose all of the equations in which the slope is -5.

• y = -5x + 2

• y = 2x - 5

• 7 - 5x = y

• -5x + y = 7

Standard Form : Ax + By = C

• Form is useful to convert to a different form.

• A, B, and C are all integers numbers.

• GCF (A, B, C)=1 --> There are no common factors in A, B, C

Slope from any equation

To find the slope from any equation, solve for y first to put the equation into slope-intercept form. THEN look at the slope.

Multiple Choice

Convert the following standard form equation to slope-intercept (y=mx+b).

• 2y = -6x + 24

• y = -3x + 12

Multiple Choice

What is the slope of the equation 3x - 4y = 8?

• 3/4

• 4/3

• -3/4

• -4/3

Parallel Lines : Slopes are the same

Perpendicular Lines : Opposite Reciprocal slopes

Use point-slope and then convert to slope-intercept.

Multiple Select

Choose all of the equations in which the slope is -2/3.

• 
  

  $$y=-\frac{2}{3}x\ -\ 8$$

• $$2x\ -\ 3y\ =\ 15$$

• $$-2x\ +\ 3y\ =\ 15$$

• $$6x\ +\ 9y\ =\ -21$$

Multiple Choice

Write the equation of the line parallel to y = 3x - 4/3 through the point (-1, 2) in slope-intercept form.

• y - 2 = 3 (x +1)

• y = 3x + 5

• y = 3x + 1

Fill in the Blank

What is the slope in the following equation?

$$y-7=\frac{1}{2}\left(x+2\right)$$  

Fill in the Blank

What is the slope of the line that is perpendicular to y = 3/4x + 5/3?

Do not type any spaces

Zero Slope

Since the slope is the number in front of x, the equation for ZERO slope looks like this: y=0x+b.

However, since any number (in this case, "x") times 0 equals 0, the equation ends up looking like this: y=b, where b is the y-intercept.

Multiple Select

Which graphs show zero slope?

Multiple Choice

What is the equation for the line shown in the graph?

• y = 4

• x = 4

• y = -4

• x = -4

Undefined Slope

A line with an undefined slope creates a vertical line. All the x-values of a vertical line are equivalent. Therefore the equation of a vertical line is x = a.

Multiple Select

Choose all the graphs that show an undefined slope.

Fill in the Blank

Type the equation for the graph shown.

Parallel Slopes

In order for two lines to be parallel, the slopes of their equations must be the same.

Multiple Select

Select the two equations that would create parallel lines.

• $$y=\frac{2}{3}x-9$$

• $$y=-\frac{3}{2}x\ +\ 6$$

• $$y=-\frac{2}{3}x-5$$

• $$y=\frac{3}{2}x+9$$

• $$y=\frac{2}{3}x-4$$

Fill in the Blank

Type the equation of the line that would be parallel to y=3x+2, but would have a y-intercept of -5. (No spaces)

Perpendicular Slope

In order for two lines to be perpendicular, the slopes of their equations must be opposite reciprocals of one another.

_{Perpendicular lines can also be made if one line has a zero slope and the other has an undefined slope.}

Multiple Select

Select the two equations that would create perpendicular lines.

• $$y=5x+7$$

• $$y=-5x-9$$

• $$y=-\frac{1}{5}x+3$$

• $$y=0.5x+2$$

• $$x=\frac{1}{5}y$$

Multiple Choice

Which equation represents the line that would be perpendicular to y = 5, and would pass through the point (-7,2)?

• $$y=5$$  

• $$x=-7$$  

• $$y=-7$$  

• $$x=2$$  

• $$y=2$$  

Introduction to Slopes and Intercepts

OBJECTIVE: Identify the sign of the slope and intercepts given a graph.

Coordinate Plane

The coordinate plane is made of two axes.

The x-Axis is the horizontal (left to right) line.

The y-Axis is the vertical (up and down) line.

Ordered Pairs

Always written in the form (x,y)

They tell us WHERE the point is on the coordinate plane.

The x-value tells us to move left or right

The y-value tells us to move up or down

Multiple Choice

What are the coordinates of Point P?

• (-3,2)

• (2,-3)

Describing the Graph

• Where it crosses the y-axis is the y-intercept.

• Where it crosses the x-axis is the x-intercept.

• How STEEP it is is called the SLOPE.

y-intercept

• Can be stated TWO WAYS

• ONE: just the number where is crosses the y-axis. In this case -2.

• TWO: the coordinates of where it crosses. In this case (0,-2).

x-intercept

• Can be stated TWO WAYS

• ONE: just the number where is crosses the x-axis. In this case 4.

• TWO: the coordinates of where it crosses. In this case (4,0).

Multiple Choice

What is the y-intercept of the line shown?

• (0,-4)

• (-4,0)

• (0,1)

• (1,0)

Multiple Choice

What is the x-intercept of the line shown?

• (0,-4)

• (-4,0)

• (0,1)

• (1,0)

Open Ended

What do these two lines have in common?

Open Ended

What is different about these two lines?

POSITIVE SLOPE

UP from left to right

NEGATIVE SLOPE

DOWN from left to right

SLOPE OF ZERO

HORIZONTAL line

UNDEFINED SLOPE

VERTICAL line

Multiple Select

Which graph(s) have a POSITIVE slope?

Multiple Select

Which graph(s) have a NEGATIVE slope?

Multiple Select

Which graph(s) have an UNDEFINED slope?

Multiple Select

Which graph(s) have a slope of ZERO?