1. Linear Functions
Chapter 5

2. Front Flap Notes Point-Slope Form: y – y1 = m(x – x1)
Slope-Intercept Form: y = mx+b
Standard Form: Ax + By = C
Slope Formula:
Parallel Lines: Equal Slopes
Perpendicular Lines: Opposite Reciprocal Slopes

3. Warmup – Rate of Change
Dixie left Austin with an odometer reading of 12,584 miles and a full tank of gas. When she stopped to buy gas in Houston, her odometer reading was 12,792 miles. She filled the tank completely with 8 gallons of gas and paid $31.12.

4. #1 – What is Slope?
Given two points on a line, slope is defined as the change in y over the change in x.
Also helpful to think of as rate of change.
Formula to Calculate
Slope is often abbreviated as m

5. #1 – Classification of Slopes

6. #1 – Classification of Slopes

7. #1 – Slope from a Graph
Calculate the slope of the line passing through
(0, 4) and (1, –2).
P1 is (0, 4) P2 is (1, –2)Identify point #1 and #2.
m = (–2 – 4)/(1 – 0) Substitute
m = –6/1 Evaluate
m = –6 Simplify

8. #1 – Slope from a Graph Calculate the slope of the line:
P1 (–4, 2) P2 is (2, 6)Identify point #1 and #2.
m = (6 – 2)/(2 – (–4)) Substitute
m = 4/ Evaluate
m = 2/ Simplify

9. #1 – Calculating Slope w/ 2 Points
Calculate the slope of a line passing through the points (–7, 9) and (2, 3).
Recall the slope formula:
Assign points and substitute into slope formula
m = (3 – 9)/(2 – (–7)) Substitute
m = –6/9 Evaluate
m = –2/3 Simplify

10. #1 – Calculating Slope w/ 2 Points
Special Cases:
(–2, 1), (–2, –4)
(5, 1), (–3, 1)

11. #1 – Practice Calculate m (slope) between each set of points.
(2, 4), (5, 13)
m = 9/3 or 3
(–1, 2), (4, –1)
m = –3/5
(–6, 5), (–6, –3)
m = undefined

12. #2 – Parallel and Perpendicular Lines
Parallel Lines
Have equal slopes
Example: y = 3x + 6 would be parallel to y = 3x – 9
Perpendicular Lines
Have opposite reciprocal slopes (opposite sign and flipped)
Example: y = –1/5x + 2 would be perpendicular to y = 5x + 6

13. #3 – Special Cases
Horizontal and Vertical Lines

14. #3a – Graph w/ Point–Slope
Graph: y – 2 = 3(x + 3)
Graph will go through point (–3, 2) and have a slope of 3.

15. #3a – Graph w/ Point–Slope
Graph: y + 4 = 2/3(x – 1)
Graph will go through point (1, –4) and have a slope of 2/3.

16. #4a – Point-Slope w/ 2 Points
Write a linear equation in point-slope form of the line passing through the points (0, 4) and (1, 6).
1st Step – Find the slope between the two points.
m = 6 – 4/1 – 0 = 2
2nd Step – Substitute slope and any known point into point-slope equation.
y – y1 = m(x – x1)
y – (6) = 2(x – (1))
Simplify and rewrite final linear equation
y – 6 = 2(x – 1) OR y – 4 = 2(x – 0)

17. #4a – Point-Slope from Graph
Write a linear equation in point-slope form of line shown.
1st Step – Identify slope of line
m = –1 – (–5)/3 – 0 = 4/3
2nd Step – Substitute slope and any
known point into point-slope equation.
y – y1 = m(x – x1)
y – (–1) = 4/3(x – (3))
Simplify and rewrite final linear equation
y + 1 = 4/3(x – 3) OR y + 5 = 4/3(x – 0)

18. #4a – Point-Slope w/ Table of Values
Write a linear equation in point-slope form of the line passing through the points in the table.
1st Step – Identify slope of line with any two points
m = 0 – 3/8 – 4 = –3/4
2nd Step – Substitute slope and any known point
into point-slope equation
y – y1 = m(x – x1)
y – (12) = –3/4(x – (–8))
Simplify and rewrite final linear equation
y – 12 = –3/4(x + 8) x y -8 12 -4 9 4 3 8

19. Warmup – Linear vs. Proportional
After 3 months of gym membership, Mr. S had paid $110. After 7 months, he had paid $190.

20. #3b – Graph w/ Slope–Intercept
Graph: y = –2x – 5
Graph will go through point (0, –5) and have a slope of –2.

21. #3b – Graph w/ Slope–Intercept
Graph: y = 3/2x + 2
Graph will go through point (0, 2) and have a slope of 3/2.

22. #4b – Slope-Intercept w/ 2 Points
Write a linear equation in slope-intercept form of the line passing through the points (0, 4) and (1, 6).
1st Step – Find the slope between the two points.
m = 6 – 4/1 – 0 = 2
2nd Step – Write equation to point-slope form.
y – y1 = m(x – x1)
y – 6 = 2(x – 1) OR y – 4 = 2(x – 0)
Simplify and rewrite final linear equation in slope-intercept
y = 2x + 4

23. #4b – Practice Write a linear equation in slope-intercept form:
(2, 5), (0, 13)
y = –4x + 13
(–2, 2), (6, –4)
y = –3/4x + 1/2
(–6, –12), (–8, –17)
y = 5/2x + 3

24. Warmup – Identify Components
Identify m (slope) and y-intercept.
y = 2/3x – 13
Identify (x1, y1) and m.
y + 2 = –1/5(x + 5)

25. What is Standard Form? Ax + By = C, where: A, B, and C are integers
Fractional or decimal coefficients must be removed by multiplication
Can be multiplied or divided by any number, given that operation is performed on both sides of equation
Can be obtained by rewriting either point-slope or slope-intercept form equations
A and B CAN’T BOTH equal 0 (one or the other can equal 0)

26. #3c – Graphing w/ Intercepts
What are Intercepts?
Intercepts are the points of intersection between a graph of a linear equation and the x or y axis.
Does every linear equation have intercepts?
Yes, every linear equation will have either one or two intercepts depending on the slope of the line.

27. #3c – Graphing w/ Intercepts
Graph using the x and y intercepts
3x + 7y = 21
To find x-intercept,
substitute 0 for y and
solve for x.
To find y-intercept,
substitute 0 for x and
solve for y.

28. #3c – Practice Find the intercepts of each equation: 3x – 3y = 6
x-intercept = 2, y-intercept = –2
4x – 2y = 10
x-intercept = 5/2, y-intercept = –5
–3x + 5y = –15
x-intercept = 5, y-intercept = –3

29. #4c – Rewriting to Standard Form
Rewrite the point-slope equation to standard form:
y – 8 = 2/3(x + 3)
Distribute: y – 8 = 2/3x + 2
Move x term: –2/3x + y – 8 = 2
Move constant: –2/3x + y = 10
Multiply common multiple: 3(–2/3x + y) = 3(10)
Simplify and rewrite: –2x + 3y = 30
Check of Equivalence using (–3, 8)
(10) – 8 = 2/3((0) + 3) –2(0) + 3(10) = 30
2 = = 30

30. #4c – Rewriting to Standard Form
Rewrite the slope-intercept equation to standard form:
y = 2/7x – 14
Move x term: –2/7x + y = –14
Multiply common multiple: 7(–2/7x + y) = 7(–14)
Simplify and rewrite: –2x + 7y = –98
Check of Equivalence using (0, –14)
(–14) = 2/7(0) – –2(0) + 7(–14) = –98
–14 = – –98 = –98

31. #4c – Equivalent Standard Form Equations
Rewrite an equivalent standard form equation to:
2x + 3y = 12
Multiply both sides by 4: 4(2x + 3y) = 4(12)
Simplify and Rewrite: 8x + 12y = 48
Check of Equivalence using (0, 4)
2(0) + 3(4) = (0) + 12(4) = 48
12 = = 48

32. #4c – Equivalent Standard Form Equations
Rewrite an equivalent standard form equation to:
1/5x + 2/3y = 5
Multiply both sides by 15: 15(1/5x + 2/3y) = 15(5)
Simplify and Rewrite: 3x + 10y = 75
Check of Equivalence using (5, 6)
1/5(5) + 2/3(6) = 5 3(5) + 10(6) = 75
5 = = 75

33. Warmup – Linear vs. Proportional
Little Timmy is 5 years old and 35 inches tall. If he continues to grow at the same rate, how tall will he be when he is 10 years old?

34. Warmup – Problem Solving
Mr. Thomas bought his first cell phone. His service is $45 per month. After 7 months he paid $464. How much did he pay for the actual cell phone?
1st Step – Substitute slope into equation.
y – y1 = 45(x – x1)
2nd Step – Substitute known coordinate pair into equation.
y – 464 = 45(x – 7)
Simplify and rewrite to slope-intercept form.
y – 464 = 45x – 315
y = 45x + 149
The y-intercept equals 149, therefore the phone cost $149.

35. Warmup – Problem Solving
After 5 months of having a cell phone, Mr. Thomas paid a total of $314. After 12 months, he paid a total of $685. Write a linear equation to model the total amount he has paid as a function of months of service.

36. Warmup – Write an Equation
Write a linear equation of the line passing through the points (2, 7) and (-1, 8) in BOTH:
Point-Slope Form
y – 7 = –1/3(x – 2) OR
y – 8 = –1/3(x + 1)
Slope-Intercept Form
y = –1/3x + 23/3 OR
y = –1/3x + 7 2/3

37. Warmup – Identify ∥ and ⊥ Slopes
y + 6 = 2/3(x – 5)
∥ = 2/3
⊥ = –3/2
y = –7x + 4
∥ = –7
⊥ = 1/7
3x + 5y = 18
∥ = –3/5
⊥ = 5/3

38. #7 – Write Parallel Equation
Write an equation of a line that passes through the point (–7, 2) and is parallel to 4x – 2y = –10.
Identify key info: ∥m = 2, (x1, y1) = (–7, 2)
Substitute: y – 2 = 2(x + 7)
IF REQUIRED, REWRITE TO SLOPE-INTERCEPT
Distribute: y – 2 = 2x + 14
Move constant: y = 2x + 16
Check using original point (–7, 2)
(2) = 2(–7) + 16
2 = 2

39. #8 – Write Perpendicular Equation
Write an equation of a line that passes through the point (9, –4) and is perpendicular to y = 3x – 6.
Identify key info: ⊥m = –1/3, (x1, y1) = (9,–4)
Substitute: y + 4 = –1/3(x – 9)
IF REQUIRED, REWRITE TO SLOPE-INTERCEPT
Distribute: y + 4 = –1/3x + 3
Move constant: y = –1/3x – 1
Check using original point (9, –4)
(–4) = –1/3(9) – 1
–4 = –4

40. Warmup – Write Equation in 3 Forms
Write the equation of the line passing through points
(-4, 1) and (-2, 5) in:
Point-Slope Form:
y – 5 = 2(x + 2) , or y – 1 = 2(x + 4)
Slope-Intercept Form:
y = 2x + 9
Standard Form:
-2x + y = 9

41. Warmup – Rewrite to Standard Form
y + 4 = –3(x + 4)
3x + y = –16
y = –3/5x – 4
3x + 5y = –20