Unit 4 Linear Functions - Weebly

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Unit 4 Linear Functions

4.1 Identifying Linear Functions 4.2 Using Intercepts 4.3 Rate of change and Slope 4.4 The Slope Formula 4.6 Slope­Intercept Form 4.7 Point­Slope Form 4.8 Line of Best Fit 4.9 Parallel/Perpendicular Lines

4.1 Identifying Linear Functions Standards: A.CED.2, A.REI.10, F.IF.5, F.IF.7, F.LE.2 Objectives: Students will be able to

Vocabulary:

Word Definition Example

Linear function

Linear equation

Identifying a Linear Function by its Graph In order for a relation to be a function, every x­value __________________________________________________________. We can determine whether or not a graph is a function, by using the __________________________________________________________. State whether each graph is a function. Explain. If the graph represents a function, is the function linear?

Example 2)

Example 3)

Identifying a Linear Function by Using Ordered Pairs In a linear function, a constant change in x corresponds to a __________________________________________________________.

Linear Function Not a Linear function

Tell whether each set of ordered pairs satisfies a linear function. Example 1) (2,4), (5,3), (8,2), (11,1)

Ex 2) (­10,10), ,(­5,4), (0,2), (5,0) Ex3) (3,5), (5,4), (7,3), (9,2), (11,1)

Ex 4) (0,5), (­2,3), (­4,1), (­6,­1), (­8,­3) Ex 5)

Standard Form of a Linear Equation

Write in Standard Form Ex 1) xy − 2 = 3 Ex 2) x − y = 5 Ex 3) x − y 4 = 2 + 5

Graphing Linear Functions Tell whether each function is linear. If so, graph the function. Ex 1) y = x + 3

Ex 2) xy = 2 Ex 3) x y = 5 − 9

Ex 4) 2 y = 1 Ex 5) y = 2x

4.2 Using Intercepts Standards: A.CED.2, A.CED.3, F.IF.2, F.IF.4, F.IF.5, F.IF.7 Objectives: Students will be able to

Vocabulary:

Word Definition Example

y­intercept

x­intercept

Finding Intercepts

To find the x­intercept, replace y with 0 and solve for x. To find the y­intercept, replace x with 0 and solve for y. Cover up method: x­intercept (cover up y term), y­intercept (cover

up x­term) Find the x­ and y­ intercepts. Ex 1) Ex 2) x y 2 3 − 2 = 1

Ex 3) Ex 4) x y 0− 3 + 5 = 3

Travel Application Ex 1) The Sandia Peak Tramway in Albuquerque, New Mexico, travels a distance of about 4500 meters to the top of Sandia Peak. Its speed is 300 meters per minute. The function f(x) = 4500­300x gives the tram’s distance in meters from the top of the peak after x minutes. Graph this function and find the intercepts. What does each intercept represent?

Ex 2) The school store sells pens for $2.00 and notebooks for $3.00. The equation describes the number of pens x and notebooks yx y 02 + 3 = 6 that you can buy for $60. ­Graph the function and find its intercepts. ­What does each intercept represent?

Graphing Linear Equations by Using Intercepts

Step 1: Find the intercepts Step 2: Graph the line using the intercepts

Use intercepts to graph the line described by each equation. Ex 1) x y2 − 4 = 8 Ex 2) y x3

2 = 4 − 21

Ex 3) x y − 2− 3 + 4 = 1 Ex 4) xy = 3

1 − 2

4.3 Rate of Change and Slope Standards: F.IF.6 Objectives: Students will be able to

Vocabulary:

Word Definition Example

rate of change

rise

sun

slope

change in dependent variable (y) Rate of change= ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­

change in independent variable (x) Real World Applications Ex 1)

Ex 2)

Finding Slope

The rise is the difference in ______________________________. The run is the difference in ______________________________. The slope of a line is the ________________________________.

Ex 1) Find the slope of the line Ex 2) Find the slope.

that contains (0,­3) and (5,­5).

Ex 3) Ex 4) (­4, 2) and (2, 0)

Ex 5) (­3,2) and (4,8) Ex 6) (­1,7) and (0,5) Finding Slopes of Horizontal and Vertical Lines

Horizontal lines have a slope of 0. Vertical lines have an undefined slope.

Find the slope of each line Ex 1) Ex 2)

4.4 The Slope Formula Standards: F.IF.6 Objectives: Students will be able to

The Slope Formula

Words Formula Example

The slope of a line is the ratio of the difference in y­values to the difference in x­values between any two different points on the line.

If are, x ) and (y , )(x1 2 1 y2 points then slope = x −x2 1

y −y2 1

If (2,­3) and (1,4) are two points on a line, then the slope is

−m = 1−24−(−3) = 7

−1 = 7

Finding Slope by Using the Slope Formula

Find the slope of the line that contains the following points Ex 1) (4,­2) and (­1,2) Ex 2) (­2,­2) and (7,­2) Ex 3) (5,­7) and (6,­4) Ex 4) , ) and ( , )( 4

357

41

52

Ex 5) (3,6) and (6,9) Ex 6) (2,7) and (4,4) Finding Slope from Graphs and Tables Find the slope of each line. Ex 1) Ex 2)

Ex 3) Ex 4)

Real World Application Ex 1) The graph shows how much water is in a reservoir at different times. Find the slope of the line. Then tell what the slope represents.

Ex 2) The graph shows the height of a plant over a period of days. Find the slope of the line. Then tell what the slope represents.

Finding Slope from an Equation

Step 1: Find the x­intercept (x­intercept, 0) Step 2: Find the y­intercept (0, y­intercept) Use the slope formula to find the slope

Ex 1) x y 0 6 − 5 = 3 Ex 2) x y 22 + 3 = 1

Ex 3) y 30 3x 5 = 1 − 1 Ex 4) y x 7 − 3 = 9 Error Analysis Two students found the slope of the line that contains (­6,3) and (2,­1). Who is incorrect? Explain the error.

4.6 Slope­Intercept Form Standards: A.CED.2, A.CED.3, F.BF.1, F.IF.6, F.IF.7, F.LE.2 Objectives: Students will be able to

Graphing by Using Slope and y­Intercept Graph each line with the given slope and y­intercept Ex 1) − and y ntercept 4 m = y − i Ex 2) and y ntercept −m = 2 − i = 3

Ex 3) and y ntercept m = 3−2 − i = 1 Ex 4) and y ntercept 3 m = 2

1 − i =

Writing Linear Equations in Slope­Intercept form

Write the equation that describes each line in slope­intercept form. Ex 1) lope and y ntercept 6 S = 3

1 − i = Ex 2) and y ntercept −m = 0 − i = 5 Ex 3) Ex 4) m = 4, (2,5) is on the line

Ex 5) − 2, y ntercept m = 1 − i = 2

−1 Ex 6)

Using Slope­Intercept Form to Graph Write each equation in slope­intercept form. Then graph the line describes by the equation. Ex 1) x y = 4 − 3 Ex 2) x y = 3

−2 + 2

Ex 3) x y3 + 2 = 8 Ex 4) x y 06 + 2 = 1

Real World Applications Ex 1) To rent a van, a moving company charges $30.00 plus $0.50 per mile. The cost as a function of the number of miles driven is show in the graph.

Ex 2) Pauline’s health club has an enrollment fee of $175 and costs $35 per month. Total cost as a function of number of membership months is shown in the graph.

4.7 Point­Slope Form Standards: A.CED.2, A.CED.3, F.BF.1, F.IF.7, F.LE.2 Objectives: Students will be able to

Writing Linear Equations in Point­Slope Form Write an equation in point­slope form.

Ex 1) lope ; (− , ) S = 25 3 0 Ex 2) lope − ; 4, ) S = 7 ( 2

Ex 3) lope ; , ) S = 2 ( 2

1 1 Ex 4) lope ; (3,− )S = 0 4 Using Point­Slope Form to Graph Graph the line described by each equation. Ex 1) (x ) y − 1 = 3 − 1 Ex 2) (x )y + 2 = 2

−1 − 3

Ex 3) − x ) y + 2 = ( − 2 Ex 4) − (x ) y + 3 = 2 − 1

Writing Linear Equations in Slope­Intercept Form Write the equation that describes each line in slope­intercept form.

Ex 1) − , (− ,− ) is on line m = 4 1 2 Ex 2) (1,­4) and (3,2) are on the line Ex 3) x­intercept = ­2 Ex 4) , (− , ) is on line m = 3

1 3 1 y­intercept = 4

Using Two Points to Find Intercepts Find the x­ and y­ intercepts Ex 1) 4, ) and (− ,− 2) ( 8 1 1 Ex 2) 2, 5) and (− ,− )( 1 4 3 Ex 3) (5,2) and (7,4) Ex 4) 5, ) and (− ,− ) ( 6 1 1 Real World Application Ex 1) The cost to place an ad in a newspaper for one week is a linear function of the number of lines in the ad. The costs for 3, 5, and 10 lines are shown. Write an equation in slope­intercept form that represents the function. Then find the cost of an ad that is 18 lines long.

Ex 2) At a different newspaper, the costs to place an ad for one week are shown. Write an equation in slope­intercept form that represents this linear function. Then find the cost of an ad that is 21 lines long.

4.8 Line of Best Fit Standards: S.ID.6, S.ID.7, S.ID.8, S.ID.9 Objectives: Students will be able to

Vocabulary:

Word Definition Example

line of best fit

linear regression

Correlation Coefficient

Linear Regression The table shows the latitudes and average temperatures of several cities. Draw a scatter plot, draw a trend line, and estimate the equation of the trend line in slope­intercept form.

*Instead of estimating the equation of the trend line, we can find the exact equation of the line using our calculators.

Step 1: Enter data in L1 (independent variables) Step 2: Enter data in L2 (dependent variables) Step 3: Press Stat → Calc → 4:LinReg(ax+b) → enter twice Step 4: Write the equation in slope­intercept form xy = m + b Ex 2) Kyle and Marcus designed a quiz to measure how much information adults retain after leaving school. The table below shows the quiz scores for several adults, matched with the number of years each person had been out of school. find an equation for a line of best fit.

Ex 3) The table shows numbers of books read by students in an English class over a summer and the students’ grades for the following semester.

a) Find an equation for a line of best fit. b) Interpret the meaning of the slope and y­intercept. c) Use your equation to predict the grade of a student who reads 15

books. Correlation Coefficient

You can find the correlation coefficient, in your calculator. Follow the same steps above, the correlation coefficient is the r.

Ex 1) What is the correlation for the example above?

4.9 Parallel/Perpendicular Lines Standards: CC.9­12, G.GPE.5, F.IF.7 Objectives: Students will be able to

Vocabulary:

Word Definition Example

Parallel Lines

Perpendicular Lines

Parallel Lines

Ex 1) Identify which lines are parallel. Ex 2) Which lines are parallel?

Ex 3) Ex 4)

Perpendicular Lines

Ex 1) Which lines are perpendicular? Ex 2)

Ex 3) Ex 4)

We can conclude that:

Lines that are parallel have _______________________________. Lines that are perpendicular have __________________________.

Writing Equations of Parallel and Perpendicular Lines Ex 1) Write an equation in slope­intercept form for the line that passes through (4,5) and is parallel to the line described by x 0y = 5 + 1 Step 1) Use the line described to find the slope

Step 2) Write the equation in point­slope form (xy − y1 = m − x1) Step 3) Write the equation in slope­intercept form xy = m + b Ex 2) Write an equation in slope­intercept form for the line that passes through (3,2) and is perpendicular to the line described by x . y = 3 − 1 Ex 3) Write an equation in slope­intercept form for the line that passes through (5,7) and is perpendicular to the line described by x . y = 5

4 − 6 Ex 4) Write an equation in slope­intercept form for the line that passes through (­5,3) and is parallel to the line described by x. y = 5

Points for Packet

4.1 Identifying Linear Functions /5

4.2 Using Intercepts /5 4.3 Rate of change and Slope /5 4.4 The Slope Formula /5 4.6 Slope­Intercept Form /5 4.7 Point­Slope Form /5 4.8 Line of Best Fit /5 4.9 Parallel/Perpendicular Lines /5 Total: /40