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2. Algebra II Linear, Exponential and Logarithmic Functions 2014-01-02 www.njctl.org

3. click on the topic to go to that section Table of Contents Linear Functions Exponential Functions Logarithmic Functions Properties of Logs Solving Logarithmic Equations e and ln Growth and Decay

4. Linear Functions Return to Table of Contents

5. Linear Functions Goals and Objectives Students will be able to analyze linear functions 
using x and y intercepts, slope and different 
forms of equations.

6. Linear Functions Why do we need this? Being able to work with and analyze lines is very 
important in the fields of Mathematics and Science. 
Many different aspects of life come together in linear 
relationships. For example, height and shoe size, 
trends in economics or time and money. Quickly, even 
these situations become non-linear, but we can still 
model some information using lines.

7. Linear Functions We will begin with a review of linear functions: a) x and y intercepts b) Slope of a line c) Different forms of lines:  i) Slope-intercept form of a line  ii) Standard form of a line  iii) Point-slope form of a line d) Horizontal and vertical lines e) Parallel and perpendicular lines f) Writing equations of lines in all three forms

8. x-int = (-6, 0) y-int = (0, 8) Linear Functions Teacher Definitions: x and y intercepts Graphically, the x-intercept is where the graph crosses the x-axis. To find it algebraically, you set y = 0 and solve for x. Graphically, the y-intercept is where the graph crosses the y-axis. To find it algebraically, you set x = 0 and solve for y. y 10 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 Find the x and y intercepts on the graph to the right. Write answers as coordinates. -10 x-int = y-int =

9. Linear Functions x-int = (5, 0) y-int = (0, 3) Find the x and y intercepts on the graph to the right. Write answers as coordinates. Teacher y x-int = y-int = 10 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

10. Linear Functions y Teacher Slope Have students then figure out the slope of these three lines. Dark Blue = -2 Green = 0 Red = -1/2 Purple = 4/3 10 An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept. Examples of lines with a y-intercept of ____ are shown on this graph. What is the difference between them (other than their color)? 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

11. Linear Functions Def: The Slope of a line is the ratio of its rise over its run. For notation, we use "m." Stress the difference and advantages 
to both methods. Teacher You can find slope two ways: y Algebraically: Graphically: count 10 8 run 6 4 rise 2 x 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 -4 -6 -8 -10

12. y Linear Functions 10 Find the slope of this line. m = 8 Teacher m = 4/3 6 rise 4 2 run x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

13. Linear Functions 1 Find the slope of the line to the right. m = -3/4 Make the rise and run bigger or 
smaller to emphasize that there 
are many places to find slope. Teacher y (Move the rise and the run to fit the slope. Make them bigger or smaller to get an accurate slope.) 10 8 6 4 2 run x 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 -4 rise -6 -8 -10

14. Linear Functions y m = 2/3 Make the rise and run bigger or 
smaller to emphasize that there 
are many places to find slope. 10 2 Find the slope of the line to the right: Teacher 8 6 (Move the rise and the run to fit the slope. Make them bigger or smaller to get an accurate slope.) 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 -4 -6 run -8 -10 rise

15. Find the slope of the line going through the following points.

16. Linear Functions 3 Find the slope of a line going through the following points: Teacher m = -8/7 (-3, 5) and (4, -3)

17. Linear Functions 4 Find the slope of the line going through the following points. Teacher m = 3/5 (0, 7) and (-5, 4)

18. Linear Functions Get ideas from students 
where slope is used in the 
"real world." Teacher Slope formula can be used to find the constant of 
change in "real world" problems. Mountain Highways Roofs Growth Distance Height And many more...

19. Linear Functions When traveling on the highway, drivers will set the cruise control 
and travel at a constant speed. This means that the distance traveled is a constant increase. The graph below represents such a trip. The car passed mile-
marker 60 at 1 hour and mile-marker 180 at 3 hours. Find the slope of the line and what it represents. Distance (miles) (3,180) (1,60) Time (hours)

20. Linear Functions 5 If a car passes mile-marker 100 in 2 hours and mile-marker 200 after 4 hours, how many miles per hour is the car traveling? Teacher m = 50mph

21. Linear Functions 6 How many meters per second is a person running if they are at 10 meters in 3 seconds and 100 meters in 15 seconds? Teacher m = 7.5 meters per second

22. Linear Functions We are going to look at three different forms of the equations of lines. Each has its advantages and disadvantages in their uses. Slope-Intercept Form Standard Form Point-Slope Form y = mx + b Ax + By = C y - y1 = m(x - x1) Advantage: Easy to find slope, y-
intercept and graph the line 
from the equation. Advantage: Easy to find intercepts and 
graph. Advantage: Can find equation or 
graph from slope and any 
point. Disadvantage: Cumbersome to 
put in another form. Disadvantage: Must manipulate it 
algebraically to find slope. Disadvantage: Must be solved for y.

23. Determine the equation of the line from the given graph. 1) Determine the y-intercept. Remember, graphically, the y-intercept is where the graph crosses the y-axis. b = -3 click 2) Graphically find the slope from any two points. click 3) Write the equation of the line using the slope-intercept form. y = mx + b y = 2x - 3 click

24. Linear Functions D y = x - 6 Ask students which form it is in: Slope-intercept Teacher 7 Which equation does 
this line represent? A y = -6x - 6 B y = -x - 6 C y = 6x + 6 D y = x - 6

25. Linear Functions A 8 Which graph 
represents the 
equation y = 3x - 2? Teacher B D line D A Line A B Line B C Line C C D Line D D

26. 9 What equation does line A represent? A Teacher B A y = 2x + 3 B B y = -2x + 3 C y = 0.5x + 3 D y = -0.5x + 3 C D

27. 10 What equation does line B represent? Teacher D A y = 2x + 3 A B B y = -2x + 3 C y = 0.5x + 3 D y = -0.5x + 3 C D

28. Linear Functions Consider the equation 4x - 3y = 6. Which form is it in? Graph it using the 
advantages of the form. It is in Standard Form. Graph 
it using the x and y 
intercepts. x-int: (3/2, 0) y-int: (0, -2) Teacher

29. Graph the equation 5x + 6y = -30 using the most appropriate method based on the form. It is in Standard Form. Graph 
it using the x and y 
intercepts. x-int: (-6, 0) y-int: (0, -5) Teacher

30. Graph the equation 3x - 5y = -10 using the most appropriate method based on the form. It is in Standard Form. Graph 
it using the x and y 
intercepts. x-int: (-10/3, 0) y-int: (0, 2) Teacher

31. Write the equation in standard form. Multiply both sides of the equation by the LCD (6). Rearrange the equation so that the x and y terms are on the same side together. Make sure that x is always positive.

32. Linear Functions 11 What is the Standard Form of: A 3x + 5y = 15 B 9x + 15y = 35 C 15x - 9y = 35 B: 9x + 15y = 35 Multiply each term by LCD: 15 Teacher D 5x - 3y = 15

33. 12 Which form is this equation in? y - 3 = 4(x + 2) A Standard Form Teacher B Slope-Intercept Form Point-Slope Form C Point-Slope Form

34. Linear Functions 13 Find y when x = 0. y - 3 = 4(x + 2) Teacher Point-Slope Form y is 11

35. Linear Functions 14 Put the following information for a line in Point-Slope form: m = -3 going through (-2, 5). Teacher C y - 5 = -3(x + 2) A y - 3 = -2(x + 5) B y - 2 = 5(x - 3) C y - 5 = -3(x + 2) D y + 2 = -3(x - 5)

36. Linear Functions 15 Put the following information for a line in Point-Slope form: m = 2 going through (1, 4). Teacher C y - 4 = 2(x - 1) A y + 4 = 2(x + 1) B y + 4 = 2(x - 1) C y - 4 = 2(x - 1) D y - 4 = 2(x + 1)

37. Linear Functions 16 Put the following information for a line in Point-Slope form: m = -2/3 going through (-4, 3). Teacher A y + 3 = -2/3(x + 4) D y - 3 = -2/3(x + 4) B y - 3 = -2/3(x - 4) C y + 3 = -2/3(x - 4) D y - 3 = -2/3(x + 4)

38. Horizontal and Vertical Lines Linear Functions Example of Horizontal line y=2 Example of a Vertical Line x=3 Vertical lines have an undefined slope. *Notice that a vertical line will "cut" the x axis and has the equation of x = n. Horizontal lines have 
a slope of 0. *Notice that a horizontal line will "cut" the y axis and has the equation of y = m.

39. Linear Functions C y = 4 Easy way to remember horizontal and vertical 
equations: Horizontal lines "cut" the y-axis at the number 
indicated. Therefore, it is y = n. Vertical lines "cut" the x-axis at the number 
indicated. Therefore, it is x = m. 17 Which equation does 
the line represent? Teacher A y = 4x B y = x + 4 C y = 4 D y = x

40. Linear Functions 18 Describe the slope of the following line: y = 4 (Think about which axis it "cuts.") Teacher B Horizontal A Vertical B Horizontal C Neither D Cannot be determined

41. Linear Functions 19 Describe the slope of the following line: 2x - 3 = 4. Teacher A  Vertical A Vertical B Horizontal C Neither D Cannot be determined

42. Parallel and Perpendicular Lines Perpendicular lineshave opposite , reciprocal 
slopes. Perpendicular lineshave opposite , reciprocal 
slopes. Parallel lines have the same slope: q(x) = x + 2 h(x) = x + 6 r(x) = x - 1 h(x) = -3x - 11 g(x) = 1/3x - 2 s(x) = x - 5

43. Linear Functions Drag the equation to 
complete the statement. Have students come to the board and drag. A) y = -1/4x - 3 B) 1/5y = x - 2 C) y = 4x - 1 D) y = -1/5x + 9 E) 6x + y = 10 F) y = 1/5x G) y = 1/6x - 6 A) y = 4x - 2 is perpendicular to _____________. B) y = -1/5x + 1 is perpendicular to _____________. C) y - 2 = -1/4(x - 3) is perpendicular to _____________. D) 5x - y = 8 is perpendicular to _____________. E) y = 1/6x is perpendicular to _____________. F) y - 9 = -5(x - .4) is perpendicular to _____________. G) y = -6(x + 2) is perpendicular to _____________. Teacher

44. How to turn an equation from point-slope to slope-intercept form. How to turn an equation from standard form to slope-intercept form. Rearrange the equation so that the y term is isolated on the right hand side. Distribute Rearrange the equation so that the y term is isolated on the right hand side. Make sure that y is always positive.

45. Are the lines 2x - 3y = 7 and 4x - 6y = 11 parallel? Turn the equations into slope-intercept form. Both slopes are 2/3. Yes the lines are parallel.

46. Are the lines 2x + 5y = 12 and 8x + 20y = 16 parallel? Turn the equations into slope-intercept form. Teacher Both slopes are 2/5. Yes the lines are parallel.

47. Linear Functions 20 Which line is perpendicular to y = -3x + 2? Teacher B A y = -3x + 1 B C y = 5 D x = 2

48. Linear Functions 21 Which line is perpendicular to y = 0? Teacher D x = 2 A y = -3x + 1 B y = x C y = 5 D x = 2

49. Linear Functions Writing Equations of Lines Slope-Intercept Form: Easiest to use if you have slope and y-intercept. If not, you will have to 
solve for b. Standard Form: Not very useful to find equations, but easy to find intercepts and graph 
with. Point-Slope Form: Easiest form to use in general. The downside is most questions will ask 
for another form. Remember the three different forms of the equations of lines. Brainstorm when each form would be easiest to use when having to write an equation of the line. Teacher Slope-Intercept Form Standard Form Point-Slope Form y = mx + b Ax + By = C y - y1 = m(x - x1)

50. Linear Functions Writing an equation in Slope- Intercept Form: From slope and one or two points: 1. Find the slope. Use either given 
information or the formula: 2. Plug the coordinates from one point 
into x and y respectfully. 3. Solve for b. 4. Write as y = mx + b. From slope and y-intercept: 1. Just plug into slope into 
m and y-intercept into b. 2. Write as y = mx + b