Quadratic Applications Section 2.1A Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 1/17/2019 10:32 AM §2.1A: Quadratic Applications

Steps for Completing the Square Put terms with variables on one side and CONSTANT to the other side Make sure the side is in DESCENDING order Standard Form, 𝑨 𝒙 𝟐 +𝑩𝒙+𝑪=𝟎 where 𝑨=𝟏 Identify the coefficient which is raised to the first power, DIVIDE the term by 2, and SQUARE the number (it will always be positive) ADD to both sides to the equation Put the equation into FACTORED form (Vertex Form) SQUARE ROOT both sides and cancel the binomial Solve for 𝒙 and check 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 1 Given 𝒚= 𝒙 𝟐 +𝟒𝒙−𝟓, convert to Vertex Form 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 2 Given 𝒚=𝟐 𝒙 𝟐 −𝟖𝒙+𝟑, convert to Vertex Form 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 2 Given 𝒚=𝟐 𝒙 𝟐 −𝟖𝒙+𝟑, convert to Vertex Form 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Your Turn Given 𝒙 𝟐 −𝟏𝟔𝒙=−𝟐, convert to Vertex Form 1/17/2019 10:32 AM §2.1A: Quadratic Applications

Using Technology to Solve When a function’s graph intersects the 𝒙-axis and the 𝒚-value is a zero, it is called a zero, solution, or root. In the real-world, it is also called the 𝒙-intercept. The maximum or minimum will establish the vertex When units are in feet, the equation is 𝒉=−𝟏𝟔 𝒕 𝟐 +𝒗𝒕+𝒄 where 𝒕 is in time and 𝒄 is initial height When units are in meters, the equation is 𝒉=−𝟒.𝟗 𝒕 𝟐 +𝒗𝒕+𝒄 where 𝒕 is in time and 𝒄 is initial height 1/17/2019 10:32 AM §2.1A: Quadratic Applications

Where does −𝟏𝟔𝒕 𝟐 and −𝟒.𝟗𝒕 𝟐 Come From? Gravity pulls objects toward the center of the earth (“down” to us) at an acceleration of 32 feet per sec2 (American measure) or 9.8 meters per sec2 (metric measure). AVERAGE acceleration per second is what is used. An object’s velocity will be greater at the end of the one-second interval than at the beginning of the interval. If acceleration at 𝒕=𝟎 is 0 and at 𝒕=𝟏 is −𝟗.𝟖, then the average is −𝟒.𝟗 in that interval. 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 3 Using the graphing calculator, identify the zeros, 𝒚-intercept, and vertex for the equation, 𝒚=𝟓 𝒙 𝟐 −𝟑𝒙−𝟒. 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4 Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . Write an equation relating to the time (𝒕) in seconds and height (𝒉) of the ball in feet. Find the height of the ball after 1.5 seconds. When does the ball reach its maximum height? What is the maximum height? When will the ball be 17 feet in the air? How long will it take for the ball to hit the ground? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4a Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . Write an equation relating to the time (𝒕) in seconds and height (𝒉) of the ball in feet. 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4b Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . b) Find the height of the ball after 1.5 seconds. 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4c Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . c) When does the ball reach its maximum height? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4d Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . d) What is the maximum height? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4e Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . e) When will the ball be 17 feet in the air? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 4f Suppose a ball is thrown upwards from a height of 5 feet with an initial velocity of 𝟑𝟐 𝒇𝒕 𝒔𝒆𝒄 . f) How long will it take for the ball to hit the ground? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 5 Johanna threw a water balloon upward at a speed of 10 m/sec while standing on the roof of a building that is 12 meters high. Write an equation relating the time (𝒕) in seconds and height (𝒉) of the balloon in meters. What was the height of the balloon after 2 seconds? When will the ball reach its maximum height? What is the maximum height? When will the ball be 5 meters in the air? Assume the balloon did not land on the roof. How long it took the balloon to reach the ground? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Your Turn Jamie threw a stone upward at 10 meters/sec while standing on a cliff 40 meters above the ground. Write an equation relating the time (𝒕) in seconds and height (𝒉) of the balloon in meters. What was the height of the balloon after 1.25 seconds? When will the ball reach its maximum height? What is the maximum height? When will the ball be 10 meters in the air? How long will it take for the stone to reach the water at the bottom of the cliff? 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 6 The path of a baseball after being hit is given by the function 𝒇 𝒙 =−𝟎.𝟎𝟎𝟑𝟐 𝒙 𝟐 +𝒙+𝟑 where 𝒇 𝒙 is the height of the baseball (in feet) and 𝒙 is the horizontal distance from home plate (in feet). What is the maximum height of the baseball? (calc) 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 6 The path of a baseball after being hit is given by the function 𝒇 𝒙 =−𝟎.𝟎𝟎𝟑𝟐 𝒙 𝟐 +𝒙+𝟑 where 𝒇 𝒙 is the height of the baseball (in feet) and 𝒙 is the horizontal distance from home plate (in feet). What is the maximum height of the baseball? (calc) 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Example 6 The path of a baseball after being hit is given by the function 𝒇 𝒙 =−𝟎.𝟎𝟎𝟑𝟐 𝒙 𝟐 +𝒙+𝟑 where 𝒇 𝒙 is the height of the baseball (in feet) and 𝒙 is the horizontal distance from home plate (in feet). What is the maximum height of the baseball? (calc) 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Your Turn The path of a punted football is given by 𝒇 𝒙 =− 𝟏𝟔 𝟐𝟎𝟐𝟓 𝒙 𝟐 + 𝟗 𝟓 𝒙+𝟏.𝟓 where 𝒇 𝒙 in the height in feet and 𝒙 is the horizontal distance (in feet) from the point at which the ball is punted. What is the maximum height of the punt? Round to 4 decimal places. 1/17/2019 10:32 AM §2.1A: Quadratic Applications

§2.1A: Quadratic Applications Assignment Worksheet 1/17/2019 10:32 AM §2.1A: Quadratic Applications