Topic 3 Mrs. Daniel- Algebra 1

Table of Contents 5.1: Understanding Linear Functions 5.2: Using Intercepts 5.3: Interpreting Rates of Change and Slope 6.1: Slope-Intercept Form 6.2: Point-Slope Form 6.3: Standard Form 6.4: Transforming Linear Functions 6.5: Comparing Properties of Linear Functions

Lesson 5.1: Understanding Linear Functions Essential Question: What is a linear function?

Linear Function Definition: Each x-value is only paired with one y- value AND the graph is a non-vertical straight line. Linear functions can not have x2 or x3 terms.

Graphing a Linear Function Determine if the function is linear Re-arrange, so y is alone Determine ordered pairs using a table of values Plot points Connect points in a line.

1. Graph: -4x + y = 11

2. Graph: 6x + y -12 X Y

Discrete vs. Continuous Function

3. Discrete or Continuous? Domain: _______________ Range: _______________

X Y Discrete or Continuous? Domain: _______________ 4. Kristoff rents a kiosk in the mall to open an umbrella stand. He pay $6 to mall owner for each umbrella he sells. The amount Kristoff pays is given by f(x) = 6x, where x is the number of umbrellas sold. Graph the function. X Y Discrete or Continuous? Domain: _______________ Range: _______________

5. A hiker is at a height of 700 meters and begins her descent 5. A hiker is at a height of 700 meters and begins her descent. She decants at a rate of 13 m/min. Which function rule describe the climber's height of elevation after t minutes?

6.

5.2: Using Intercepts Essential Questions: How can you identify and use intercepts in linear relationships?

Vocab: Intercepts X-intercept: y-coordinate of the point where the graph intercepts the y-axis. The x-coordinate of this point is always zero. Y-intercept: x- coordinate of the point where the graph intercepts the x-axis. The y-coordinate of the point is always zero.

x-intercept: ___________ y-intercept: ________ y-intercept: ________

x-intercept: ___________ y-intercept: ________ y-intercept: ________

Find the Intercepts Algebraically To find x-intercept, replace y with zero and solve for x. To find y-intercept, replace x with zero and solve for y.

5. Find Intercepts: -5x + 6y = 60 x-intercept y-intercept

6. Find Intercepts: 8x + 7y = 28 x-intercept y-intercept

7. Find Intercepts: -6x - 8y = 24 x-intercept y-intercept

8. The temperate is an experiment is increased at a constant rate over a period of time until the temperature reaches 0°C. The equation y = 5 2 x -70 give the temperate y in digress Celsius x hours after the experiment begins. Find the x and t intercepts.

Graph in Standard Form How to: Rewrite equation in standard form (Ax + By = C), if need. Plug in zero for x and solve for y-intercept. Plug in zero for y and solve for x-intercept. Plot both points. Connect with line.

9. Graph using the Intercepts: 18y = 12x + 108

Use the intercepts to Graph 10. 11.

Use the intercepts to Graph 12. 13.

14.

15. Graph using the intercepts 15. Graph using the intercepts. The air temperature is -6C at sunrise and rises 3C every hours for several hours. The air temperature after x hours is represented by the function f(x)= 3x – 6.

5.3: Interpreting Average Rate of Change and Slope Essential Question: How can you relate the average rate of change and slope in a linear equation?

Average Rate of Change

2.

What is Slope? Slope: describes the steepness or incline of a line. A higher slope value indicates a steeper incline. Slopes can be positive, negative, zero or undefined. Slope is abbreviated with “m” Average rate of change is same idea as slope, but for non-linear functions.

Determining Slope Graphically We can count the rise and run on a graph to determine slope.

Slope

1. Find the Slope.

2. 3.

5. 4.

Slope Formula

6. Find the Slope.

7. Find the Slope.

8. Find the Slope.

9. Find and interpret the slope & y-intercept

10. Slope: Interpret:

11.

12. Slope: Interpret:

13.

14.

Lesson Check Quiz: 3. 1. 2.

6.1: Slope Intercept Form Essential Question: How can you represent a linear function in a way that reveals in slopes and y-intercept?

Slope Intercept Form

1. Write in Slope-Intercept… The line passes through (0, 5) and (2, 13). Write the equation of the line in slope-intercept form. a. Find the slope (m). b. Sub m, x-cord, y-cord into the equation (y = mx + b) to solve for b. c. Sub m and b into the equation.

2. Write in Slope-Intercept… The line passes through (3, 5) and (2, -1). Write the equation of the line in slope-intercept form. a. Find the slope (m). b. Sub m, x-cord, y-cord into the equation (y = mx + b) to solve for b. c. Sub m and b into the equation.

3. Find the slope- intercept form Slope is 3, and (2, 5) is on the line

4. Find the slope- intercept form The line passes through (0,-3) and (2,13).

Graphing the Slope Intercept Form of a Line

Graphing the Equation of a Line Start at “b”. Move up/down Move left/right Repeat

7. Graph: y = 5x -4

8. Graph: 2x + 6y = 6

9. Graph: 2x + 3y = 6

10. Graph: 2x + y = 4

11. A chairlift descends from a mountain top to pick up skiers at the bottom. The height in feet of the chairlift is a linear function of the time in minutes since it begins descending as shown in the graph. Slope: Interpret: Y-intercept: Equation:

12. A local club charges an initial membership fee as well as a monthly cost. The cost C in dollars is a linear function of the number of months of membership. Slope: Interpret: Y-intercept: Equation:

Lesson 6.2: Point-Slope Form Essential Question: How can you represent a linear equation in a way that it reveals its point and slope on its graph?

Point Slope Equation

Write the Point-Slope Equation Slope is 3.5 and (-3, 2) is on the line. Slope is 0 and (-2, -1) is on the line.

5. Solve using any method Paul wants to place an ad in a newspaper. The newspaper charges $10 for the first 2 lines of text and $3 for each additional line of text. Paul’s ad is 8 lines long. How much will the ad cost?

6. An animal shelter asks all volunteers to take a training session and then to volunteer for one shift each week. Each shift is the same number of hours. The table shows the number of hours Joan and her friend Miguel worked over several weeks. Another friend, Lili, plans to volunteer for 24 weeks over the next year. How many hours will Lili volunteer?

7. A roller skating rink offers a special rate for birthday parties 7. A roller skating rink offers a special rate for birthday parties. One the same day, a party of 10 skaters cost $107 and a party for 15 skaters costs $137. How much would a party for 12 skaters cost?

Lesson 6.3: Standard Form Essential Question: How can you write a linear equation in Standard Form given properties of the line including its slopes and points on the line?

Comparing Forms of Linear Equations

Write in Standard Form… Example: Slope is 4 and (-2, 1) is on the line. How to: Plug into point-slope. Rearrange into Standard Form.

1. Write in Standard Form… Slope is 5 and (-2, 4) is on the line.

Let’s Practice… 2. 3.

4. Write in Standard Form… (-2, -1) and (0, 4) are on the line. How to: Find slope. Identify b. Then plug in m and b to slope-intercept form and rearrange to standard form. OR Plug into point-slope and rearrange to standard form.

5. Write in Standard Form… (5, 2) and (3, -6) are on the line.

Let’s Practice… 6. 7.

8. Write an Equation in Standard Form A tank is filling up with water at a rate of 3 gallons per minute. The tank already had 3 gallons in it before it started being filled.

9. Write an Equation in Standard Form A hot tub filled with 440 gallons of water is being drained. After 1.5 hours, the amount of water had decreased to 320 gallons.

Graph

11. Write an Equation in Standard Form

Write an Equation in Standard Form 12. 13.

14. Write an equation in Standard Form…

Lesson 6.4: Transforming Linear Functions Essential Question: What are the ways in which you can transform the graph of a linear function?

What happens when…. The gym lowers the one time fee to join? The gym increased the monthly fee?

C. Once a year the gym offers a special in which the one-time fee for joining is waived for new members. What impact does this special offer have on the graph? D. Suppose the gym increases its one-time joining fee and decreases its monthly membership fee. Does this have any impact on the domain of the function?

Changes to… Words Visually Slope (m) Intercept (b)

2.

3.

Transformation Rule Applying translations to a function DO NOT change the SHAPE of the function, only its LOCATION .

4.

Compare the x and y coordinates of the vertices. Find h and k. Transform Functions: Find the coordinates of the vertices of the functions f(x) (green) and g(x) (red). Compare the x and y coordinates of the vertices. Find h and k. Describe the transformation(s). Write the transformation rule. 5. f: V (__,__) g: V’(__,__) h = ______ k = _______ Transformation: ____________________________________________ Rule: _________________

6. f: V (__,__) g: V’(__,__) 7. f: V (__,__) g: V’(__,__) h = ______ k = _______ h = ______ k = _______ Transformation: _________________________________________ Transformation(s): _________________________________________ Rule: _________________ Rule: _________________

Compare the x and y coordinates of the vertices. Find h and k. Transform Functions: Find the coordinates of the vertices of the functions f(x) (green) and g(x) (red). Compare the x and y coordinates of the vertices. Find h and k. Describe the transformation(s). Write the transformation rule. 8. f: V (__,__) g: V’(__,__) h = ______ k = _______ Transformation: ____________________________________________ Rule: _________________

9. f: V (__,__) g: V’(__,__) 10. f: V (__,__) g: V’(__,__) h = ______ k = _______ h = ______ k = _______ Transformation: _________________________________________ Transformation(s): _________________________________________ Rule: _________________ Rule: _________________

Let’s Practice…

Lesson 6.5: Comparing Properties of Linear Functions Essential Question: How can you compare linear functions that are represented in different ways?

Comparing 2 Functions The domain of each function is the set of all real numbers x such that 5≤x≤8. The table shows some ordered pairs for f(x). The function g(x) is defined by the rule g(x) = 3x + 7. What is the f(x) function rule? f(x) initial value? g(x) initial value? f(x) range? g(x) range?

Comparing 2 Functions The domain of each function is the set of all real numbers x such that 6≤x≤10. The table show some ordered pairs for f(x). The function g(x) is defined by the rule g(x) = 5x + 11. What is the f(x) function rule? f(x) initial value? g(x) initial value? f(x) range? g(x) range?

1. Write a function rule for each, and then compare their domain, range, slope and y-intercept. A rainstorm in Austin lasted 3.5 hours, during which time it rained at a steady rate of 4.5 mm per hour. The graph shows the amount of rain that fell during the same rainstorm in Dallas, D(t) as a function of time. Austin Dallas Function Rule Domain Range Slope Y-intercept

2. Write a function rule for each, and then compare their domain, range, slope and y-intercept. The first group of hikers hiked at steady rate of 6.5 km per hour for 4 hours. The graph shows the 2nd group of hikers. 1st Group 2nd Group Function Rule Domain Range Slope Y-intercept

3.

4.

6.