SLOPE AND RATE OF CHANGEEQUATIONS OF LINES

Section 3.4 & 3.5

Slope and Rate of Change

Section 3.4 & 3.5

Find the slope of a line given two points on the line.

Find the slope of a line given its equation.Find the slopes of vertical and horizontal lines.Compare the slopes of parallel and perpendicular

lines.Use the slope-intercept form to write an equation

of a line.Use the slope-intercept form to graph a linear

equation.Find equations of vertical and horizontal lines.

Finding the Slope of a Line Given Two Points of the Line

Section 3.4 & 3.5

A key feature of a line is its slant or steepness. This measure is called the slope of the line.

Slope is the ratio of the vertical change to the horizontal change between two points as we move along the line.

Can be found on a graph by counting the vertical rise and horizontal run.

Finding the Slope of a Line Given Two Points of the Line

Find the slope of the line between1. A to B

Down 3 -3 Right 3 +3

B to A Up 3 +3 Left 3 -3

The direction traveled does not matter, the slope will be the same either way.

2. A and C

3. B and C

Section 3.4 & 3.5

31

3

4

72

31

3

Summary of Slope

Section 3.4 & 3.5

The slope of a line is always interpreted by reading the line as it moves from left to right.

Upward Line

Positive Slope

m > 0

Vertical Line

Undefined Slope

m does not exist

Horizontal Line

Zero Slope

m = 0

Downward Line

Negative Slope

m < 0

(1, 0)

(4, 3)

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Finding the Slope of a Line Given Two Points of the Line

Consider the points (1, 0) and (4, 3) in the coordinate plane.

Count the “rise” and the “run” to determine the slope of the line between the points.

3

3

3 – 0

4 – 1

Section 3.4 & 3.5

Finding the Slope of a Line Given Two Points of the Line

Section 3.4 & 3.5

The Slope Formula Given two points and , where , the

slope of the line connecting the two points is given by the formula

Find the slope of the line connecting the given points.1. (6, -2) and (5, 5)2.

1 1,x y 2 2,x y 1 2x x

2 1

2 1

y ym

x x

x y

-5 7

-2 5

1 3

4 1

723

Using the Slope-Intercept Form to Graph an Equation

Graph the line passing through the point (-2, 5) with a slope of -3.1. To graph, plot the

known point. Slope is the “rise” and

“run” needed to get from one point to another along a line.

2. Follow the slope to reach a second point.

3. Connect the points with a straight line.

Section 3.4 & 3.5

33

1

Using the Slope-Intercept Form to Graph an Equation

Through the given point, draw a line with the given slope.

1. (3, 2) m = 4

3. (2, -7) m = 0

2. (-2, -1) m = 2/7

4. x-int: -3, m = -1/2

Section 3.4 & 3.5

Finding the Slope of a Line Given Its Equation

Section 3.4 & 3.5

Slope-Intercept Form When a linear equation in two variables is written in slope-

intercept form, , m is the slope of the line and (0, b) is the y-intercept of the line.

Find the slope and y-intercept of the line.1.

2. Equation must be written in y = mx + b

3. All horizontal lines have a slope of zero.

4. All vertical lines have undefined slope.

59 3y x

2x

1y

7 2 8x y

59 ; (0,3)m

72 ; (0, 4)m

0; (0,1)m

;m undefined does not exist

y mx b

Using the Slope-Intercept Form to Write an Equation

Slope-Intercept Form When a linear equation in two variables is written in

slope-intercept form, , m is the slope of the line and (0, b) is the y-intercept of the line.

Find the equation of the line with y-intercept (0, 7) and slope of ½.

Find the equation of the line through the points (1, 3) and (0, 4)

y mx b

12 7y x

4y x

Section 3.4 & 3.5

Finding Equations of Vertical and Horizontal Lines

Find an equation of the horizontal line through the point (1, 4).

Find an equation of the vertical line through the point (1, 4).

4y

1x

Section 3.4 & 3.5

Using the Slope-Intercept Form to Graph an Equation

If an equation is in slope-intercept form, both the slope and a point on the line are known.

Slope-Intercept Form

Section 3.4 & 3.5

y mx b slope y-intercept

(0, b)

Using the Slope-Intercept Form to Graph an Equation

To graph an equation in slope-intercept form y = mx + b:

1. Plot the y-intercept (0, b).

2. Plot a second point by rising the number of units indicated by the numerator of the slope then running the number of units indicated by the denominator of the slope, m.

3. Draw a straight line through the points.

Find the slope and y-intercept of each line, then graph.5.

6.

Section 3.4 & 3.5

3 4y x

13 2y x

Using the Slope-Intercept Form to Graph an Equation

Determine the slope and y-intercept, then graph.

7. 2x + y = 8

9. 4x – 3y = 9

8. 4y = -8x

10. y = 8

Section 3.4 & 3.5

2 8y x

43 3y x

2y x

Using the Slope-Intercept Form to Graph an Equation

Determine the slope and y-intercept, then graph.

11. 2x – y = 4

13. 3x – 4y = 4

12. x = 4

14. x = 3/2 y

Section 3.4 & 3.5

2 4y x

34 1y x

23y x

Slopes of Parallel and Perpendicular Lines

Parallel

Parallel lines have the same slope.

Perpendicular

Perpendicular lines have slopes that are negative reciprocals, or a product of -1.

Section 3.4 & 3.5

Slopes of Parallel and Perpendicular Lines

Section 3.4

Determine if the pair of lines is parallel, perpendicular, or neither.1.

2.

3.

5 1

5 10

y x

x y

2 3 21

6 4 2

x y

y x

11

2 11

x y

x y

11 2 55;m m

perpendicular

2 21 23 3;m m

parallel

1 21; 2m m

neither