Unit 5: Quadratic Functions LESSON #2: THE PARABOLA APPLICATIONS AND WORD PROBLEMS INVERSE OF A QUADRATIC FUNCTION DO NOW: Review from Lesson #1 (a)using the graph shown to the right, determine the equation of the axis of symmetry. = (b)what are the coordinates of the turning point? Classify the point as a maximum or minimum. @ (, ) (c) Identify all of the intercepts of the graph. @, (, ) @, (d)identify the interval(s) on which the graph is increasing and decreasing. :, De :, (e)find the average rate of change on the interval.. = = = with Parabolas 1
Ex 1) The table below represents a type of bacteria that doubles every 36 hours. A Petri dish starts out with 12 of these bacteria. Calculate and interpret the average rate of change over the interval [2,5]. = Every 3 days, =. there is an Each day, there is an increase of approx. increase of 91 bacteria. 30.33 bacteria. Ex 2) Calculate the average rate of change of = + over the interval,. Answer: = with Parabolas 2
Applications of Parabolas Ex 1) A toy rocket is launched from an 89m tall platform. The equation for the object s height, ( ), at a time seconds after launch is =. + +. (a)how long does it take for the toy rocket to reach its maximum height? [Round to nearest hundredth] (b)what is the maximum height of the toy rocket? [Round to nearest meter] (c)at what time does the toy rocket return to the ground? [Round to nearest second] (d) When will the projectile reach 50m above the ground? [Round to nearest second] Answers: (a). secs (b) m(c) secs(d) secs with Parabolas 3
Ex 2) Alex is standing on a hill 80 feet high. He throws a baseball upward with an initial velocity of 64 feet per second. The height of the ball, ( ), in terms of the time seconds since the ball was thrown is = + +. (a)how long does it take for the ball to reach its maximum height? (b)what is the maximum height of the ball? (c) What is the height of the ball after 3 seconds? (d)at what time does the ball hit the ground? Answers: (a) secs (b) ft (c) ft (d) secs Ex 3) The weekly profit function in dollars of a small business that produces fruit jams is =. +, where is the number of jars of jam produced and sold. (a)find the number of jars of jam that should be produced to maximize the weekly profit. (b)find the maximum profit. Answers: (a) jars (b) $ with Parabolas 4
Ex 4) Tom throws a football with an initial velocity of 80 feet per second from a height of 6 feet above the ground. equation, =. + + gives the path of the ball, where ( )is the height and is the horizontal distance the ball travels. (a) What is the maximum height reached by the football? (b) A receiver catches the ball 3 feet above the ground. How far has the ball traveled horizontally when the receiver catches it? Answers: (a) ft (b) ft Using a graphing calculator, graph the pairs of equations on the same graph. Sketch your results. A B C y y y = = = x 2 + 3 x 3 What do you notice about the graphs? x 3 Graphs B and C create the inverse of Graph A Is the original parabola a function? Yes. Is the inverse parabola a function? No. with Parabolas 5
Algebraically, find the inverse of = +. Recall: To find an inverse, switch and. Then, solve for. What does the ± translate to graphically? Top and Bottom of the inverse parabola! = + = ± = =± What happens, graphically, if the ±is NOT included in the equation of the inverse? = Is this inverse parabola a function? Yes! Only the Top half of the inverse parabola! with Parabolas 6
Recall: The inverse of a function may not always be a function. The original function must be a one-to-one function to guarantee that its inverse will also be a function. Is there any way to create an inverse function, algebraically? Recall: One-to-one function each and value is used only once and passes the horizontal line test! Yes! Algebraically, find the inverse function of = +. Recall: To find an inverse, switch and. Then, solve for. How can we find the inverse FUNCTION? Do NOT introduce the ±when finding the square root! = + = = Creates only the Top half of the inverse parabola! with Parabolas 7
Ex 1) Given the function below, find the equation for. Switch and. Then, solve for. = = = = Inverse MUST be a function! = = Ex 1) Given the function below, find the equation for. Switch and. Then, solve for. = + + = + + Inverse MUST be a function! How do we solve for? Complete the Square! = + + + = + + =( + ) = + = with Parabolas 8