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**Chapter 9: Quadratic Equations and Functions**

Lesson 4: Quadratics and Projectiles Mrs. Parziale

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Vocabulary: object projectile: is an ______________ that is thrown, dropped, or launched, and then proceeds with no additional force on its own.

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**c Rules or Properties: y-intercept up down minimum maximum**

In the equation y = ax2 + bx + c the initial value or starting point is the _____________ which is the _______ value. The coefficient a determines whether the graph opens ________ or ______________. The vertex is a _______________ on a graph that opens upward and a _________________ on a graph that opens downward. y-intercept c up down minimum maximum

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**Properties vertex time height zero**

In a projectile problem, the object being projected reaches a maximum height at its _________. The x-coordinate represents _______________ the projectile travels. The y-coordinate represents _______________ the projectile is from the ground. When the projectile reaches the ground, the y-coordinate is ________. vertex time height zero

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**General Formula for the height of a Projectile over Time:**

where h = height (feet), t = time, v = initial velocity, s = initial height (ft) where h = height (meters), t = time, v = initial velocity, s = initial height (meters)

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Example 1: A ball is thrown from an initial height of 6 feet with an initial upward velocity of 32 feet per second. a) Write a formula describing the height of the object (in feet) after t seconds. b) How high will the ball be ½ second after it is thrown? c) What is the maximum height this ball reaches? (vertex – use the MAXIMUM function on calculator) d) When does the ball reach the ground? (x-intercepts – use the ZEROS function on calculator)

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**Example 2: An object is dropped from an initial height of 90 meters. **

What is the initial velocity? Write a formula describing the height of the object (in meters) after t seconds. After how many seconds does the object hit the ground? Set the equation equal to zero and solve for t. What is the maximum height of the object?

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Example 3: Suppose a ball is thrown upward with an initial velocity of 22 meters per second from an initial height of 2 meters. Write a formula describing the height of the object (in meters) after t seconds. What is the y-intercept? What are the x-intercepts? What do the x-intercepts imply? Estimate when the ball is 20 meters high. (graph a horizontal line y=20 and use the INTERSECT function on the calculator)

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Closure What is the formula used to find the height of a projectile over time? What does the “c” value represent? How do you find the x-intercepts? What do the x-intercepts represent? Setup this situation: A ball is batted with an initial upward velocity of 26 meters from an initial height of 1 meter. When is the ball 5 meters high?

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Example 1 Divide by 2 to make the coefficient of x 2 equal to 1. Add 8 to both sides. Add [½·2] 2 = 1 to both sides to “Complete the Square”

Example 1 Divide by 2 to make the coefficient of x 2 equal to 1. Add 8 to both sides. Add [½·2] 2 = 1 to both sides to “Complete the Square”

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