1. Linear Functions and Slope-Intercept Form Lesson 2-3
Algebra 2
Linear Functions and Slope-Intercept Form
Lesson 2-3

2. Goals Goal Rubric To graph linear equations.
To write equations of lines.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Essential Question
Big Idea: Equivalence
What is slope-intercept form?

4. Vocabulary Slope Linear Function Linear Equation y-Intercept
x-Intercept
Slope-Intercept Form

5. Rate of Change change in dependent variable (y) rate of change =
Rate of Change – a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.
The rates of change for a set of data may vary or they may be constant.
change in dependent variable (y)
rate of change =
change in independent variable (x)

6. Rates of Change on a Graph
To show rates of change on a graph, plot the data points and connect them with line segments.
Data in a graph is always read from left to right.
Remember!

7. Definition
The constant rate of change of a line is called the slope of the line.
Slope – Is a measure of the steepness of a line.
The steeper the line, the greater the slope.
Notation: m stands for slope.
For any two points, the slope of the line connecting them can be determined by dividing how much the line rises vertically by how much it runs horizontally.

8. Slope
run
rise

9. Slope

10. Example: Slope from a Graph
Find the slope of the line.
Run –9
Begin at one point and count vertically to fine the rise.
(–6, 5) • •
Rise –3
Run 9
Then count horizontally to the second point to find the run.
Rise 3
(3, 2)
It does not matter which point you start with. The slope is the same.

11. Your Turn:
Find the slope of the line that contains (0, –3) and (5, –5).
Begin at one point and count vertically to find rise.
Then count horizontally to the second point to find the run.
Run –5
It does not matter which point you start with. The slope is the same. •
Rise –2
Rise 2 •
Run 5

12. Your Turn: slope = = The slope of the line is .
Find the slope of the line.
(5, 4)
Begin at one point and count vertically to find the rise.
(1, 2)
Then count horizontally to the second point to find the run.
slope = = 2 4 1
The slope of the line is . 1 2

13. Your Turn: slope = = 1 The slope of the line is 1.
Find the slope of the line.
Begin at one point and count vertically to find the rise.
(3, 2)
Then count horizontally to the second point to find the run.
(–1, –2)
slope = = 1 4
The slope of the line is 1.

14. vertical change horizontal change
Slope Formula
Recall that lines have constant slope. For a line on the coordinate plane, slope is the following ratio:
vertical change horizontal change
change in y change in x =

15. Slope Formula y2 – y1 x2 – x1 slope =
If you know any two points on a line, you can find the slope of the line without graphing. The slope of a line through the points (x1, y1) and (x2, y2) is as follows:
y2 – y1
x2 – x1
slope =
When finding slope using the ratio above, it does not matter which point you choose for (x1, y1) and which point you choose for (x2, y2).

16. Slope Formula

17. Example:
Find the slope of the line that passes through (–2, –3) and (4, 6).
Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6). =
y2 – y1
x2 – x1
6 – (–3)
4 – (–2)
Substitute 6 for y2, –3 for y1,
4 for x2, and –2 for x1.
6 + 3
4 + 2 = 9 6 = 3 2 =
Simplify.
The slope of the line that passes through (–2, –3) and (4, 6) is . 3 2

18. Example:
Find the slope of the line that passes through (1, 3) and (2, 1).
Let (x1, y1) be (1, 3) and (x2, y2) be (2, 1). =
y2 – y1
x2 – x1
1 – 3
2 – 1
Substitute 1 for y2, 3 for y1,
2 for x2, and 1 for x1. -2 1 =
= –2
Simplify.
The slope of the line that passes through (1, 3) and (2, 1) is –2.

19. Example:
Find the slope of the line that passes through (3, –2) and (1, –2).
Let (x1, y1) be (3, –2) and (x2, y2) be (1, –2). =
y2 – y1
x2 – x1
–2 – (–2)
1 – 3
Substitute -2 for y2, -2 for y1,
1 for x2, and 3 for x1.
-2 + 2
1 – 3 =
Rewrite subtraction as addition of the opposite.
= 0 –2 =
The slope of the line that passes through (3, –2) and (1, –2) is 0.

20. Your Turn:
Find the slope of the line that passes through (–4, –6) and (2, 3).
Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3). =
y2 – y1
x2 – x1
3 – (–6)
2 – (–4)
Substitute 3 for y2, –6 for y1,
2 for x2, and –4 for x1. 9 6 = 3 2 =
The slope of the line that passes through (–4, –6) and (2, 3) is . 3 2

21. Your Turn:
Find the slope of the line that passes through (2, 4) and (3, 1).
Let (x1, y1) be (2, 4) and (x2, y2) be (3, 1). =
y2 – y1
x2 – x1
1 – 4
3 – 2
Substitute 1 for y2, 4 for y1,
3 for x2, and 2 for x1. -3 1 =
= –3
Simplify.
The slope of the line that passes through (2, 4) and (3, 1) is –3.

22. Your Turn:
Find the slope of the line that passes through (3, –2) and (1, –4).
Let (x1, y1) be (3, –2) and (x2, y2) be (1, –4). =
y2 – y1
x2 – x1
–4 – (–2)
1 – 3
Substitute -4 for y2, -2 for y1,
1 for x2, and 3 for x1.
–4 + 2
1 – 3 =
= 1 –2 =
Simplify.
The slope of the line that passes through (3, –2) and (1, –4) is –2.

23. Example: Slopes of Horizontal and Vertical Lines
Find the slope of each line. A. B.
You cannot divide by 0
The slope is undefined.
The slope is 0.

24. Your Turn: Find the slope of each line. A. B. The slope is undefined.
You cannot divide by 0.
The slope is undefined.
The slope is 0.

25. Slopes
As shown in the previous examples, slope can be positive, negative, zero or undefined. You can tell which of these is the case by looking at a graph of a line–you do not need to calculate the slope.

26. Example:
Tell whether the slope of each line is positive, negative, zero or undefined. A. B.
The line rises from left to right.
The line falls from left to right.
The slope is positive.
The slope is negative.

27. Your Turn:
Tell whether the slope of each line is positive, negative, zero or undefined. a. b.
The line is vertical.
The line rises from left to right.
The slope is positive.
The slope is undefined.

28. Comparing Slopes

29. Definition
Family of Functions - is a set of functions whose graphs have basic characteristics in common.
Example: All linear functions form a family because all of their graphs are the same basic shape, a line.
Linear Function – a function whose graph is a line.
Parent Function – is the simplest or basic function within a family of functions.
Linear Parent Function – is y = x or f(x) = x.

30. Definition
Linear Equation - is an equation that models a linear function.
In a linear equation the variables are raised to the 1st power only.
Example:
Linear equation; y = 2x + 3.
Not a linear equation; y = x2 or y = 2x.
The graph of a linear equation contains all the ordered pairs that are solutions of the equation.

31. Definition:
Recall from geometry that two points determine a line. Often the easiest points to find are the points where a line crosses the axes.
The y-intercept is the point where the line crosses the x-axis.
Example: (0, 2)
The x-intercept is the point where the line crosses the y-axis.
Example: (3, 0)

32. Slope-Intercept Form of a Linear Equation
If you know the slope of a line and the y-intercept, you can write an equation that describes the line.
Step 1 If a line has a slope of m and the y-intercept is b, then (0, b) is on the line. Substitute these values into the slope formula.

33. Slope-Intercept Form of a Linear Equation
Step 2 Solve for y:
Simplify the denominator.
Multiply both sides by x.
+b b
mx = y – b
Add b to both sides.
mx + b = y, or y = mx + b

34. Slope-Intercept Form of a Linear Equation
y = mx + b
slope
y-intercept
Any linear equation can be written in slope-intercept form by solving for y and simplifying. In this form, you can immediately see the slope and y-intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

35. Finding the Slope and y-intercept of a Line
Procedure for finding the slope and y-intercept of a line.
Rewrite the equation of the line in slope-intercept form by solving for y.
The coefficient of x, is the slope and the constant term, is the y-coordinate of the y-intercept.
y = mx + b
Slope (always comes in front of x!)
(0, b) y – intercept (where the line crosses the y-axis)

36. Example: Write the equation of the line in slope-intercept form.
slope = ; y-intercept = (0, 4)
y = mx + b
Substitute the given values for m and b.
y = x + 4
Simplify if necessary.

37. Example:
Write the equation that describes the line in slope-intercept form.
slope = –9; y-intercept = (0, )
y = mx + b
Substitute the given values for m and b.
y = –9x +
Simplify if necessary.

38. Your Turn:
Write the equation that describes the line in slope-intercept form.
slope = 3; y-intercept = (0, )
y = mx + b
Substitute the given values for m and b.
Simplify if necessary.

39. Your Turn:
Write the equation that describes the line in slope-intercept form.
slope = ; y-intercept = (0, –6)
y = mx + b
Substitute the given values for m and b.
Simplify if necessary.

40. Example:
Write the equation in slope-intercept form and find the slope and y-intercept. 2x – 3y + 6 = 0.
Procedure We need to solve for y.
This is the given equation.
2x – 3y + 6 = 0
To isolate the y-term, add 3 y on both sides.
2x + 6 = 3y
Reverse the two sides. (This step is optional.)
3y = 2x + 6
Divide both sides by 3.
The slope is 2/3 and the y-intercept is (0, 2).

41. Example:
Write the equation in slope-intercept form and find the slope and y-intercept.
6x – 3y = 18
y = mx + b
Solve for y.
y = 2x - 6
2x – 3y = 18
slope = 2
y-intercept = (0, - 6)
– 3y = 18 – 6x
y = x
y = 2x - 6

42. Example:
Write the function in slope-intercept form and find the slope and y-intercept.
Solve for y first.
(4)
(4)
Multiply both sides by 4.
Isolate the y-term by subtracting 4x from each side.
4x + 3y = 24
3y = -4x + 24
Divide both sides by 3.
y = x + 8
The y-intercept is (0, 8) and slope

43. Your Turn: 1) 2y + 12 = 6x 2) 3y – x = 12 Solve for y. Solve for y.
Write the equation in slope-intercept form and find the slope and y-intercept.
1) 2y + 12 = 6x
2) 3y – x = 12
Solve for y.
Solve for y.
2y = 6x – 12
3y = x + 12
y = 3x – 6
y = 1/3 x +4
y = mx + b
y = mx + b
y = 3x – 6
y = 1/3 x + 4
slope = 3
y-intercept = (0, –6)
slope = 1/3
y-intercept = (0, 4)

44. Your Turn:
Write the equation in slope-intercept form and find the slope and y-intercept.

45. Example: Writing an Equation From a Graph
Find the linear equation of the line shown on the graph.
y-intercept = 4 y • •
Rise = –2
Step 1 Pick two points on the line and determine the slope of the line from the graph. • •
m =
Run = 5
Step 2 Find the y-intercept from the graph.
b = 4
Step 3 Write the linear equation in slope-intercept form (y = mx+ b).

46. Your Turn: y = –x + 2 y = 2x – 2 Write the equation of the line shown.B. A.
y = –x + 2
y = 2x – 2

47. Steps for Graphing y = mx + b
You can use the slope and y-intercept from a linear equation in slope-intercept form to graph a line
Graphing y = mx + b by Using the Slope and y-Intercept
Write the equation in slope-intercept form.
Plot the y-intercept on the y-axis. This is the point (0, b).
Obtain a second point using the slope, m. Write m as a fraction, and use rise over run starting at the y-intercept to plot this point.
Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.

48. Example: Graph the line given the slope and y-intercept.•
Rise = –2 •
Step 1 The y-intercept is 4, so the line contains (0, 4). Plot (0, 4). • •
Step 2 Slope = Count 2 units down and 5 units right from (0, 4) and plot another point.
Run = 5
Step 3 Draw the line through the two points.

49. Example: Graph the line given the slope and y-intercept.
slope = 4; y-intercept =
Run = 1
Rise = 4
Step 1 The y-intercept is , so the line contains (0, ). Plot (0, ). •
Step 2 Slope = Count 4 units up and 1 unit right from (0, ) and plot another point. •
Step 3 Draw the line through the two points.

50. Your Turn: Graph the line given the slope and y-intercept.
slope = 2, y-intercept = –3
Step 1 The y-intercept is –3, so the line contains (0, –3). Plot (0, –3).
Run = 1
Rise = 2
Step 2 Slope =
Count 2 units up and 1 unit right from (0, –3) and plot another point. • •
Step 3 Draw the line through the two points.

51. Your Turn: Graph each line given the slope and y-intercept.
slope = , y-intercept = 1
Step 1 The y-intercept is 1, so the line contains (0, 1). Plot (0, 1).
Rise = –2 •
Step 2 Slope = Count 2 units down and 3 units right from (0, 1) and plot another point. •
Run = 3
Step 3 Draw the line through the two points.

52. Example:
Write the function –4x + y = –1 in slope-intercept form. Then graph the function.
Solve for y first.
–4x + y = –1
+4x x
Add 4x to both sides.
y = 4x – 1
The line has y-intercept –1 and slope 4, which is Plot the point (0, –1). Then move up 4 and right 1 to find other points.

53. Your Turn:
Run 3
y-intercept
b = -1
Rise 2
slope

54. Your Turn:
Run 2
y-intercept
Rise
b = 3 -1
slope

55. Your Turn:
Write the function 2x – y = 9 in slope-intercept form. Then graph the function.
Solve for y first.
2x – y = 9
–2x –2x
Add –2x to both sides.
–y = –2x + 9
y = 2x – 9
Multiply both sides by –1.
The line has y-intercept –9 and slope 2, which is . Plot the point (0, –9). Then move up 2 and right 1 to find other points.

56. Your Turn:
Write the function 5x = 15y + 30 in slope-intercept form. Then graph the function.
Solve for y first.
5x = 15y + 30
– –30
Subtract 30 from both sides.
5x – 30 = 15y
Divide both sides by 15.
The line has y-intercept –2 and slope . Plot the point (0, –2). Then move up 1 and right 3 to find other points.

57. Essential Question Big Idea: Equivalence What is slope-intercept form?
Slope-intercept form is a special form of a linear function. Rewriting the equation of a line in slope-intercept form is useful for graphing the line.

58. Assignment:
Section 2-3, Pg 85 – 87; #1 – 6 all, 8 – 54 even.