unit 4 linear functions - weebly
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Name:________________ Block: _______
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