Linear Equations and Functions

Transcription

1 Linear Equations and Functions Finding the Slope and Equation of a Line. ans x y y x b c c b Finding Slope.... undefined undefined (, -) Finding the Equation of a Line in Slope-Intercept Form. y = x +. y x.6. y = -x +

2 . y x. y x y x 7. y = -x 0 8. y = x + 9. y x 0. y x. y x 7. y = x. y = -x +. y = x +0. y x 6. 9 y x 7. y = - 8. y = y = b 0. x = a Standard Form of a Line. y x 6. y x. m, b (0,). Answers will vary. Parallel: two lines that never intersect, lines with the same slope. Perpendicular: two lines that intersect at 90 (or right angle),. Standard Form. x + y =. x y =. x y =. y x. 9 y x 6. y x 7. (, 0), (0, ) 8.,0, (0, -8) 9. 6,0, (0, -) 0. x y =. x y = 0. x y = -. x y = 0. x + y =. 8x + y = A C y x B B 7. A, C m b B B 8. There are infinitely many possibilities. One possible answer is x y = y = 0x + 800; 880

3 Finding the Equation of Parallel Lines. y = x +. y = -. y x. x =. y x 6. y = -x y x 8. y x y x 9. y = x y x. yes. no. no. yes Finding the Equation of Perpendicular Lines. y = -x +. x = -6. y = x 7. y = -9. y = -x y x 7. y = 6x 8 8. y x 9. 7 y x 0. y = -6x + 8. y = x +. perpendicular. parallel. perpendicular. neither Graphing Lines. y = x +. y x. y x. a) (, 0) and (0, -) b) (, 0) and 0,

4 Graph a Line in Slope-Intercept Form. ans ans ans ans ans ans ans ans ans ans ans-00-. ans-00-

5 . (-6, ) ans-00-. (a, b). $7 Graph a Line in Standard Form. ans-00-. ans-00-. ans ans ans ans-00-9

6 7. ans ans ans ans-00-. ans-00-. ans-00-. For standard form, students might say it is easier to find the intercepts to graph the line. However, answers could vary. Some might like the consistency of doing the problems in the same manner, regardless of what form it is in.. Because this is a vertical line, it cannot be changed into slope intercept form. Students should recognize that the - is the x-intercept and the only intercept. So, students should use the intercepts to graph this line. Relations and Functions.. y x. y = x + 6. y = x. They are perpendicular to each other. ans-00-0

7 Defining Relations and Functions. yes. no. no. yes. no 6. yes 7. no 8. yes 9. no 0. yes. yes. no. no. yes. no 6. yes 7. yes 8. yes 9. yes 0. no, a vertical line would touch another vertical line (from the Vertical Line Test) in infinitely many places.. All lines, except vertical lines, are functions. Finding the Domain and Range of Functions. Domain: x {, -, 7, 0}, Range: y {6,, -, 9}. Not a function.. Not a function.. Domain: x {-, -6, 0, 8}, Range: y {,, 7, }. Domain: x, Range: y 6. Domain: x, Range: y 7. Domain: x, Range: y (,8] 8. Domain: x, Range: y 9. Domain: x, Range: y [, ] 0. Domain: x (, ) [, ), Range: y. Domain: x (,] [6, ), Range: y (,6]. Domain: x (, 7) (,] [, ), Range: y (,0) (,] [, ) Graphing Linear Inequalities in Two Variables -.. x ans x 0 ans x ans-00-0 ans-00-0

8 Testing Solutions for Linear Inequalities in Two Variables. all. C, D. D. A, C. D 6. A, C 7. C 8. A, C 9. no 0. yes. Possible answers: (, ), (, 0), (, -). Possible answers: (, ), (-, ), (-, 0) Graphing Linear Inequalities in Two Variables. ans ans ans ans ans ans ans ans ans-00-

9 0. ans-00-. ans-00-. ans-00-. y > -x. x. y -x 6 6. y > x Graphing Absolute Value Equations. x = 7, -. x =, -7. x = 0, -6 Graphing Basic Absolute Value Functions. v(-6, 0), Domain: x, Range: y [0, ). v(, 0), Domain: x, Range: y [0, ). v(0, ), Domain: x, Range: y (,]. v(0, -), Domain: x, Range: y [, ). v(-, 7), Domain: x, Range: y (,7] 6. v(, -6), Domain: x, Range: y [ 6, ) 7. v(0, 0), Domain: x, Range: y [0, ) 8. v(0, 0), Domain: x, Range: y [0, ) 9. v(0, 0), Domain: x, Range: y [0, ) 0. upside down. (h, k). all real numbers. narrower. wider. (9, 7)

10 Using the General Form and the Graphing Calculator. The first and third graphs are exactly the same. The only difference between those and the second function is that it is flipped upside down. Because the negative sign is inside the absolute value on the third function, it become positive, making the graph look exactly like the first. All three graphs are narrower than the parent graph.. Both graphs are wider than the parent function. The first graph is shifted over one unit to the left. The second graph is shifted up one unit.. Yes, they do all produce the same graph. When the two is inside the absolute value, it doesn t matter if it is positive or negative, the answer will always be positive. In the first function, the two is outside the graph, but because it is positive, it will be no different than the other two functions.. In general, you could say that if the vertex is (h, k) and the equation is y a x h k, the range is: y [ k, ) if a is positive and y (, k] if a is negative.. v(, ), Domain: x, Range: y [, ) 6. v(-, 0), Domain: x, Range: y (,0] 7. v(0, ), Domain: x, Range: y [, ) 8. v(-, -), Domain: x, Range: y [, ) 9. v(7, 0), Domain: x, Range: y (,0] 0. v(8, 6), Domain: x, Range: y (,6]. v,0, Domain: x, Range: y (,0]. v,, Domain: x, Range: y,. v, 7, Domain: x, Range: 7 y,

11 . Domain: x, Range: y [, ). x, Range: y [0, ) ans ans When x ( ] [, ), the ranges and graphs are exactly the same. What happens between - and is what changes. This is because in the original function, what is between - and is below the x-axis, making the range negative. Because the second function is the absolute value, the range cannot be negative. Therefore, it is like we took everything that was below the x-axis in the first function and folded it above the x-axis in the second function. Analyzing Scatterplots. Mean: 0, Median: 0, Mode:.. a) (-6, ) b) (9, ) ans c) (, 6) Plotting a Scatterplot and Finding the Equation of Best Fit. negative. none. positive. y = x. y = -x + 0

12 6. 7. y = 9x $ ans y = -00x home runs in 0 ans *Answers may vary slightly for 7, 8, 0, and. Finding the Equation of Best Fit using a Graphing Calculator. y = 9.x y = -88.6x y = 0.00x y = 0.x y = 0.x Answers may vary. One possible answer could be advances in medicine (medications and treatments) have affected the life expectancy such that people are living longer.