Chapter 2 Linear Relations and Functions
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1 Chapter Linear Relations and Functions I. Relations and Functions A. Definitions 1. Relation. Domain the variable ( ) 3. Range the variable ( ). Function a) A relationship between ( ) and ( ). b) The output ( ) depends on the input ( ). c) A in which each element in the is mapped to and onl element in the. B. Eamples 1. Determine whether each relation is a function. a) Mappings: b) Tables: c) Ordered Pairs: {( 5,),(,5),(0,7),(0,9) }. Graphs: 1) ) 3) Note: Vertical Line Test Honors Algebra Chapter Page 1
2 C. Function Values and Notation 1. Function Notation a) = 3 8 vs. f ( ) = 3 8 b) Function notation has the advantage of clearl identifing the dependent variable f ( ), formerl known as, while at the same time telling ou is the independent variable and that the function itself is called f. c) Function notation allows ou to be less word. Instead of asking What is the value of that corresponds to =? ou can ask What is f ( )? :. Eamples 3 a) If f ( ) = 3, find f (). b) If h( ) = , find h (1.6). c) If f =, find ( ) 3 ( ) 3 f t. d) Etra: If g a =, find g( b + 3). ( ) a 6 Homework: Worksheet Section.1 Honors Algebra Chapter Page
3 II. Linear Equations A. Standard Form: a + b = c, where a, b, and c are integers. B. Intercepts 1. -intercept a) Occurs when the line crosses the. b) The equals 0. -intercept a) Occurs when the line crosses the. b) The equals 0 C. Linear equations: Note: When variables other than and are used, assume that the letter coming first in the alphabet represents the or horizontal ais. D. Eamples: 1. Write in standard form: 1 3 =. Find the intercepts and graph: + = 0 Homework: Worksheet Section. III. Slope A. Definition: B. Formula: C. Eample: Find the slope of a line that passes through the points (1,-3) and (-1,6). D. Tpes of Slopes: Homework: Worksheet Section.3 Honors Algebra Chapter Page 3
4 IV. Slope Intercept Form/Point Slope Form A. = m + b where m is the slope and b is the -intercept. B. ( ) = m( ) where (, ) 1 1 is a given point and m is the slope. 1 1 C. Find the equation of a line with the following description: 1. Slope of 3 and goes through the point of (-1,6).. Goes through the points (,-1) and (5,-8). 3. Goes through the points (3,-1) and (8,-1). D. Parallel and Perpendicular Lines 1. Parallel Slopes are.. Perpendicular Slopes are an. 3. Eamples a) Find an equation of a line that is parallel to = + 5 and goes through the point (7,1). b) Find an equation of a line that is parallel to 3 5 = 7 and goes through the point (-,5). c) Find an equation of a line that is perpendicular to = + 5 and goes through the point (,1). Homework: Worksheet. Honors Algebra Chapter Page
5 V. Modeling Real-World Data (Manuall) Tpes of correlation 1. Positive correlation. Negative correlation 3. No correlation A equation can be determined b using a process to the one used to find the of a. The procedure Eamples 1. The table below shows the approimate percent of students who sent applications to two colleges in various ears since Draw a scatter plot, best fit line, and find an equation that would predict the approimate percent of students who sent applications to two colleges in various ears since Predict the percent of students who will send applications to two colleges in 010. What is the dependent and independent variable and wh? What tpe of correlation does this data have? Safet The table below shows the approimate percent of drivers who wear seat belts in various ears since 199. What is the dependent and independent variable and wh? Draw a scatter plot and a best fit line. What tpe of correlation does this data have? Find an equation that would predict the approimate percent of drivers who wear seatbelts since 199. What do the slope and -intercept indicate? Predict the percent of drivers who will be wearing seat belts in Homework: p all, 1-8 all, 30 Honors Algebra Chapter Page 5
6 Scatter Plots and Regression (prediction) Lines 1. Entering Data a. EDIT Edit To clear a List: 1. Go to the top of list. b.. Note: Do not delete lists. Enter the -values in L1 and -values in L c. QUIT In order to get deleted lists back ou must do the following: SetUpEditor This should restore our lists.. Graphing Data We do not normall do this step, unless asked to see how well the data fits. a. Clear all equations from that do not pertain to the problem at hand. b. STAT PLOT c. Select on of the Stat Plots d. Make sure the following is selected. Note: If our - or -values are in a different list location ou can alwas change the XList: and YList: locations tping in the new list location. To select put cursor on top of item and hit You can change our mark to our preference. e. QUIT f. ZoomStat Honors Algebra Chapter Page 6
7 3. Finding the Prediction Line (Model) a. Calc LinReg(a+b) b. After LinReg(a+b) ou will need to tpe the list he -values are located (usuall L1) flowed b a comma and the list where the -values are located (usuall L) followed b another comma and then the location ou want to store the equation (usuall Y1). After all is tped in hit New wa: Y1 Old wa: Y1 thru Y9 can be located b doing the following: Y-VARS Function c. You should get the following. a is our slope and b is our -intercept. For this result the equation is = Using the model to predict If ou want to predict what will happen when is 5 New wa: Y1 (5) Old wa: Y-VARS Y1(5) Honors Algebra Chapter Page 7
8 VI. Modeling Real-World Data (Using a Calculator) Correlation Coefficient ( r ): Shows how well data are modeled b a function. Measures: When r is close to 1, When r = 0, When r Eamples 1. The table shows the median income of U.S. families for the period Use a graphing calculator to make a scatter plot of the data. Find an equation for and graph a line of regression. Then use the equation to predict the median income in 015. Use a graphing calculator to make a scatter plot of the data. Find an equation for and graph a line of regression. Then use the equation to predict the attendance in 015. What is the correlation coefficient? What does this value tell us about our model?. The table shows the winning times for an annual dirt bike race for the period Use a graphing calculator to make a scatter plot of the data. Find and graph a line of regression. Then use the function to predict the winning time in 015. Homework: p all Honors Algebra Chapter Page 8
9 VII. Special Functions 1. Piecewise Function. Absolute Value Function: 3. Greatest Integer Function: ( integer not greater than ) = 3 Eample = or = or Eamples: Graph: 1, if 3 f ( ) = 1, if > 3 Find the Domain and Range. + 1, if > 1 f ( ) = 3, if 1 Find the Domain and Range. Pscholog: One pschologist charges for counseling sessions at the rate of $85 per hour or an fraction thereof. Draw a graph that represents this solution = = + f ( ) = 3 Find the Domain and Range. Find the Domain and Range. Homework: Worksheet Section.6 Honors Algebra Chapter Page 9
10 VIII. Parent Functions and Transformations Parent Graphs A. Constant Function: B. Identit Function: C. Absolute Value function: D. Quadratic Function: Transformations A. Translation 1. Horizontal:. Vertical: B. Reflection 1. Reflection over the -ais:. Reflection over the -ais: C. Dilation 1. Stretched verticall:. Compressed verticall: D. Eamples 1. Describe the translation in ( 1) Then graph the function. = +.. Describe the translation in =. Then graph the function. 3. Describe the reflection in =. Then graph the function.. Describe what transformations to the following function 1 f ( ) = +. Then graph the function. Homework: p all, 9, all, 1-19 all, 1, 8, 33, all, 5-58 all, 60 Honors Algebra Chapter Page 10
11 IX. Linear Inequalities Boundar Lines or. -- < or > Eamples: Is (,) a solution to 3 < 6? Graph: < < One tutoring compan advertises that it specializes in helping students who have a combined score on the SAT that is 900 or less. Write an inequalit to describe the combined scores of students who are prospective tutoring clients. Let represent the verbal score and the math score Homework: Worksheet Section.8 Honors Algebra Chapter Page 11
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