Chapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3.
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1 Capter Functions and Graps Section. Ceck Point Exercises. Te slope of te line y x+ is. y y m( x x y ( x ( y ( x+ point-slope y x+ 6 y x+ slope-intercept. a. Write te equation in slope-intercept form: x+ y y x+ y x+ Te slope of tis line is tus te slope of any line perpendicular to tis line is... a. Use m and te point (, 6 to write te equation. y y m( x x y ( 6 ( x ( y+ 6 ( x+ y+ 6 x+ 6 x+ y x y general form Cange in y m. Cange in x Te slope indicates tat te number of U.S. men living alone increased at a rate of. million eac year. Te rate of cange is. million men per year. c. f( x f( x x x f x x f x 7 x ( ( 8 f x x f x x ( ( ( 8 ( f( x f( x f( f(... Te average rate of cange in te drug's concentration between our and ours is. mg per ml per our.. x x 6. a. c. s( ( s( ( 6 6 feet per second s( ( s(. (. 9 9 feet per second. s( ( s(. ( feet per second. Concept and Vocabulary Ceck.. te same.. ;. ;. y; x f( x f( x x x s(6 s( 6 76 Copyrigt Pearson Education, Inc.
2 Section. More on Slope Exercise Set.. Since L is parallel to y x, we know it will ave slope m. We are given tat it passes troug (,. We use te slope and point to write te equation in point-slope form. y y m( x x y ( x Solve for y to obtain slope-intercept form. y ( x y x 8 y x 6 In function notation, te equation of te line is f x x ( 6.. L will ave slope m. Using te point and te slope, we ave y ( x. Solve for y to obtain slope-intercept form. y x+ 6 y x+ f x x+ (. Since L is perpendicular to y x, we know it will ave slope m. We are given tat it passes troug (,. We use te slope and point to write te equation in point-slope form. y y m( x x y ( x Solve for y to obtain slope-intercept form. y ( x y x+ y x+ In function notation, te equation of te line is f ( x x+.. L will ave slope m. Te line passes troug (,. Use te slope and point to write te equation in point-slope form. y ( x ( y ( x+ Solve for y to obtain slope-intercept form. y x+ y x+ + y x+ f ( x x+. m since te line is parallel to y x+ ; x 8, y ; point-slope form: y + (x + 8 slope-intercept form: y + x y x 6. m since te line is parallel to y x+ ; x, y 7 ; point-slope form: y + 7 (x + slope-intercept form: y+ 7 x y x 7 7. m since te line is perpendicular to y x+ 6; x, y ; point-slope form: y + (x slope-intercept form: y+ x+ y x m since te line is perpendicular to x, y ; point-slope form: y ( x+ slope-intercept form: y x y x y x+ 7 ; Copyrigt Pearson Education, Inc. 77
3 Capter Functions and Graps 9. x y 7 y x+ 7 7 y x Te slope of te given line is, som since te lines are parallel. point-slope form: y ( x+ general form: x y+. x y y x+ y x Te slope of te given line is, so m since te lines are parallel. point-slope form: y ( x+ general form: x y+ 9. x y y x+ y x Te slope of te given line is, so m since te lines are perpendicular. point-slope form: y+ 7 ( x general form: x+ y. x+ 7y 7y x+ y x+ 7 7 Te slope of te given line is, so m 7 since te 7 lines are perpendicular. point-slope form: y + 9 7(x general form: 7x y a ( (9 6 ( ( 6 6 ( c. d.. a. s( ( 9 s( ( feet per second s( ( 9 s(. ( feet per second. s( ( 9 s(. ( feet per second. s( ( 9 s(. ( feet per second. s( ( 8 s( ( feet per second s( ( 8 s(. ( feet per second. 78 Copyrigt Pearson Education, Inc.
4 Section. More on Slope c. d. s( ( 8 s(. ( feet per second. s( ( 8 s(. ( feet per second.. Since te line is perpendicular to x 6 wic is a vertical line, we know te grap of f is a orizontal line wit slope. Te grap of f passes troug (,, so te equation of f is f ( x.. Since te line is perpendicular to x wic is a vertical line, we know te grap of f is a orizontal line wit slope. Te grap of f passes troug (,6, so te equation of f is f ( x 6.. First we need to find te equation of te line wit x intercept of and y intercept of. Tis line will pass troug (, and (,. We use tese points to find te slope. m Since te grap of f is perpendicular to tis line, it will ave slope m. Use te point ( 6, and te slope equation of te line. y y m( x x y x 6 y ( x+ 6 y x y x+ f ( x x+ ( ( to find te. First we need to find te equation of te line wit x intercept of and y intercept of 9. Tis line will pass troug (, and (, 9. We use tese points to find te slope. m 9 9 Since te grap of f is perpendicular to tis line, it will ave slope m. Use te point (, 6 and te slope equation of te line. y y m( x x y 6 x y 6 ( x+ y 6 x y x+ f ( x x+ ( ( to find te. First put te equation x y in slopeintercept form. x y y x+ y x Te equation of f will ave slope since it is perpendicular to te line above and te same y intercept. So te equation of f is f ( x x. 6. First put te equation x y 6 in slope-intercept form. x y 6 y x+ 6 y x 6 Te equation of f will ave slope since it is perpendicular to te line above and te same y intercept 6. So te equation of f is f ( x x 6. Copyrigt Pearson Education, Inc. 79
5 Capter Functions and Graps 7. px (.x+ 8. px (.x a.. a m Tere was an average increase of approximately 7 discarges per year m 6 Tere was an average decrease of approximately discarges per year. f x x x x ( f (.( ( + 6( f (.( ( + 6( m Tis overestimates by discarges per year. f x x x x ( f (.(7 (7 + 6( f (.( ( + 6( m 96 7 Tis underestimates te decrease by discarges per year.. 8. Answers will vary. 9. y x+ y x a. Te lines are perpendicular because teir slopes are negative reciprocals of eac oter. Tis is verified because product of teir slopes is. Te lines do not appear to be perpendicular. c. Te lines appear to be perpendicular. Te calculator screen is rectangular and does not ave te same widt and eigt. Tis causes te scale of te x axis to differ from te scale on te y axis despite using te same scale in te window settings. In part (b, tis causes te lines not to appear perpendicular wen indeed tey are. Te zoom square feature compensates for tis and in part (c, te lines appear to be perpendicular.. makes sense. makes sense. does not make sense; Explanations will vary. Sample explanation: Slopes can be used for segments of te grap.. makes sense. Write Ax + By + C in slope-intercept form. Ax + By + C By Ax C By Ax C B B B A C y x B B A Te slope of te given line is. B Te slope of any line perpendicular to Ax + By + C is B A.. Te slope of te line containing (, and (, ( as slope m Solve Ax + y for y to obtain slope-intercept form. Ax + y y Ax+ So te slope of tis line is A. Tis line is perpendicular to te line above so its slope is. Terefore, A so A Copyrigt Pearson Education, Inc.
6 Mid-Capter Ceck Point 6. a. 8. a. 7. a. c. Te grap in part (b is te grap in part (a sifted down units. c. Te grap in part (b is te grap in part (a reflected across te y-axis. Mid-Capter Ceck Point. Te relation is not a function. Te domain is {, }. Te range is { 6,,6}.. Te relation is a function. Te domain is {,,}. Te range is {, }.. Te relation is a function. Te domain is { x x < }. Te range is { y y }.. Te relation is not a function. Te domain is { x < x }. Te range is { y y }. c. Te grap in part (b is te grap in part (a sifted to te rigt units.. Te relation is not a function. Te domain is {,,,,}. Te range is {,,,}. 6. Te relation is a function. Te domain is { x x }. Te range is { y y }. 7. x + y y x + For eac value of x, tere is one and only one value for y, so te equation defines y as a function of x. Copyrigt Pearson Education, Inc. 8
![Eac value of x corresponds to exactly one value of y. 6. 7.. Domain: (,. Range: (,].](/docs-images/86/94722463/images/7-6.jpg "x-intercepts: 6 and 8.. y-intercept:. increasing: (,. decreasing: (, 6. x 9. 7. f ( 8. f ( 9.")
7 Capter Functions and Graps 8. x+ y y x y ± x Since tere are values of x tat give more tan one value for y (for example, if x, ten y ± ±, te equation does not define y as a function of x. 9. No vertical line intersects te grap in more tan one point. Eac value of x corresponds to exactly one value of y Domain: (,. Range: (,]. x-intercepts: 6 and 8.. y-intercept:. increasing: (,. decreasing: (, 6. x f ( 8. f ( 9. f ( 7 and f (. f ( 6 and f (.. ( 6,. f ( is negative.. neiter; f( x x and f( x x.. f( x f( x f( f( x x ( +. 8 Copyrigt Pearson Education, Inc.
8 Mid-Capter Ceck Point.. 7. a. f( x ( x x x x neiter; f( x x and f( x x f( x+ f( x ( x+ + ( x+ ( x + x x x + x+ + x x+ x + ( x + x +. y x y x 8. if t Cx ( +.( t if t > a. C ( C ( +.(. y y 9. y y m( x x y x ( ( y ( x+ y x 8 y x f( x x 6.. Cange in y ( 6 m Cange in x ( y y m( x x ( x y y x y x f( x x Copyrigt Pearson Education, Inc. 8
9 Capter Functions and Graps. x y y x+ y x Te slope of te given line is, and te lines are parallel, so m. y y m( x x y ( ( x y+ x 9 y x f( x x. x y y x+ y x + y x f( x f( x f( f(. x x ( ( ( ( ( ( + Section.6 Ceck Point Exercises. Sift up vertically units.. Sift to te rigt units. Te slope of te given line is, and te lines are perpendicular, so m. y y m( x x y ( ( x ( y+ x y x f( x x Cange in y (. m Cange in x 7 Cange in y 6 m Cange in x ( Te slope of te lines are equal tus te lines are parallel.. Sift to te rigt unit and down units.. Reflect about te x-axis.. a. Cange in y 6 6 m.6 Cange in x 8 8. Reflect about te y-axis. For eac minute of brisk walking, te percentage of patients wit depression in remission increased by.6%. Te rate of cange is.6% per minute of brisk walking. 8 Copyrigt Pearson Education, Inc.
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