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Intercepts An Introduction Copyright 2014 Scott Storla The y-intercept of a function is the ordered pair when x is 0. An x-intercept of a function is any ordered pair when y is 0. Although the x-intercept and the y-intercept are ordered pairs, in practice 0, 8 people often refer to just the x-coordinate of the x intercept or just the y-coordinate of the y-intercept. 6, 0 Copyright 2014 Scott Storla Copyright 2014 Scott Storla Time traveling in car (hours) Distance from home (miles) 0 0 288 288 2.25 172 4.5 81 6 6 0 Copyright 2014 Scott Storla What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? What is the specific meaning of any x-intercepts? Copyright 2014 Scott Storla Intercepts An Introduction Copyright 2014 Scott Storla Applying Intercepts Graphs and Tables Copyright 2014 Scott Storla C a rs W a sh e d 0 4 6 8 12 P ro fit (d o lla rs) 60 20 0 20 60 What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? What is the specific meaning of any x-intercepts? Copyright 2014 Scott Storla What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? Why won’t there be any x-intercepts? Copyright 2014 Scott Storla What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? What is the specific meaning of each x-intercept? Copyright 2014 Scott Storla What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? Why won’t there be any x-intercepts? Copyright 2014 Scott Storla What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? What is the specific meaning of each x-intercept? Copyright 2014 Scott Storla Applying Intercepts Graphs and Tables Copyright 2014 Scott Storla Graph the data and extend the line to estimate the intercepts. Use 0 to 20 for x values and 0 to 1000 for y values. Discuss the meaning of any intercept or explain why the intercept doesn’t exist. M o n th s sin ce co u ch w as bought R e sa le v a lu e (D o lla rs) 4 600 5 550 10 300 Copyright 2014 Scott Storla Graph the data and extend the line to estimate the intercepts. Use 0 to 20 for x values and 0 to 100 for y values. Discuss the meaning of any intercept or explain why the intercept doesn’t exist. M o n th s sin ce co u ch w as bought R e sa le v a lu e (D o lla rs) 4 600 5 550 10 300 Copyright 2014 Scott Storla Graph the data and extend the line to estimate the intercepts. Discuss the meaning of any intercept or explain why the intercept doesn’t exist. A grade school is putting on a carnival. When people enter they buy tokens which they use to play the carnival games. Scale the x-axis from 0 to 2000 and the y-axis from -400 to 400. T o k e n s so ld P ro fit m a d e ($ ) 400 100 700 50 1000 200 Copyright 2014 Scott Storla Graph the data and extend the line to estimate the intercepts. Discuss the meaning of any intercept or explain why the intercept doesn’t exist. Scale the xaxis from 0 to 80 and the y-axis from 0 to 80. M in u te s p la yin g th e slo t m a ch in e M o n e y le ft (D o lla rs) 10 64 25 4 7 .5 0 40 31 Copyright 2014 Scott Storla Applying Intercepts Equations Part 1 Copyright 2014 Scott Storla To find the y-intercept of an equation substitute 0 for x and solve for y. To find the x-intercept(s) of an equation substitute 0 for y and solve for x. The function y 6 .5 x 2 0 returns y, the profit (in dollars) made from a car wash, given x, the number of cars that have been washed. W h a t's th e sp e cific m e a n in g o f th e y-in te rce p t? W h a t's th e sp e cific m e a n in g o f th e x-in te rce p t? y 6 . 5 x 20 y 6 . 5 x 20 y 6 . 5 0 20 0 6 . 5 x 20 y 20 3 . 077 x Before any cars are washed the loss is $20. To make up the $20 loss (your profit is $0) you need to wash 4 cars. Copyright 2014 Scott Storla The function y 2 x 13 estimates the percent of one-person households in the U.S., y, using x, the year since 1960. What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercept? Why won’t there be any x-intercepts? y 2 x 13 y 2 x 13 O per y 2 0 13 In v ^2 y 0 13 2 2 y 13 13 13 In 1960 around 13% of households were one-person households. 0 2 x 13 0 2 x 13 13 13 0 2 4 2 .2 5 1 3 13 2 x 2 2 6 .5 6 .5 2 0 18 . 09 ... x 42 . 25 x Copyright 2014 Scott Storla 0 2 6 .5 1 3 x 2 You can find y, the income you make on a product (in $10,000) by supplying x, the price you plan to charge for the product (in dollars) using the function y 5( x 4 ) 2 80 . What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercepts? What is the specific meaning of the x-intercepts? y 5( x 4 ) y 5( 0 4 ) 2 2 80 80 y 5 1 6 8 0 y 80 80 y 0 Setting the price to $0 leads to an income of $0. O p e r In v 4 4 ^2 5 5 80 80 Setting the income to $0 leads to a price of $0 or $8. Copyright 2014 Scott Storla y 5( x 4 ) 0 5( x 4 ) 2 2 80 80 8 0 5( x 4 ) 16 x 4 2 2 16 x 4 44 x x 8 or x 0 Applying Intercepts Equations Part 1 Copyright 2014 Scott Storla Applying Intercepts Equations Part 2 Copyright 2014 Scott Storla y The function y 3 1,5 0 0 e 0 . 15 x predicts the current value for a hybrid car of age x. What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercepts? What is the specific meaning of the x-intercepts? y 3 1, 5 0 0 e y 3 1, 5 0 0 e 0 . 15 x 0 . 15 ( 0 ) y 3 1, 5 0 0 e 0 y 3 1, 5 0 0 (1 ) y 31, 500 e O per In v 0 .1 5 0 .1 5 e^ ln 3 1, 5 0 0 3 1, 5 0 0 0 31, 500 e 0 The value doesn’t go to $0. The car initially cost $31,500. e Copyright 2014 Scott Storla 0 e 0 0 . 15 x 3 1, 5 0 0 e 3 1, 5 0 0 y 3 1, 5 0 0 0 . 15 x 0 . 15 x 3 1, 5 0 0 0 . 15 x ln 0 ln e 0 . 15 x e rro r 0 . 1 5 x x y The number of years since 1985 can be used to predict the pounds of beef consumed per person per year in the United States using the formula y 5 . 5 ln ( x 1) 79 . What is the general meaning of the y-intercept? What is the specific meaning of the y-intercept? What is the general meaning of any x-intercepts? What is the specific meaning of the x-intercepts? y 5 . 5 ln ( x 1) 7 9 y 5 . 5 ln ( 0 1) 7 9 y 5 . 5 ln (1) 7 9 y 5 .5 0 7 9 y 79 In 1985 approximately 79 pounds of beef was consumed per person. y 5 . 5 ln ( x 1) 79 O per In v 1 1 ln e^ 5 .5 5 .5 79 79 0 5 . 5 ln ( x 1) 79 79 79 79 5 . 5 ln ( x 1) 5 .5 5 .5 79 5 . 5 ln ( x 1) The pounds of beef consumed will not drop to 0. Copyright 2014 Scott Storla e 79 5 . 5 e ln ( x 1 ) 1730007 x 1 Applying Intercepts Equations Part 2 Copyright 2014 Scott Storla