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NOTES FOR WS 5-8 SOLVING QUADRATIC APPLICATIONS

2. Football Problem: When a football is punted, it goes up into the air, reaches a maximum altitude,

then comes back down. Find a quadratic model for this situation.

Let t = number of seconds that have elapsed since the ball was punted

Let d = number of feet the ball is above the ground

a. When the ball was kicked it was 4 feet above the ground. One second later, it was 28 feet above

the ground. Two seconds after it was kicked, it was 20 feet up. Write the equation expressing d in

terms of t.

Step 1: Write the ordered pairs. (0,4) (1,28) (2,20)

Step 2: Write the standard form of the quadratic function.

d(t) = at2 + bt + c

Step 3: Substitute each ordered pair into the standard form of the quadratic function

(0,4) 4 = a(0)2 + b(0) + c This gives 0a + 0b + 1c = 4.

(1, 28) 28 = a(1)2 + b(1) + c This gives 1a + 1b + 1c = 28.

(2, 20) 20 = a(2)2 + b(2) + c This gives 4a + 2b + 1c = 20.

Step 4: Write a matrix equation using the 3 x 3 system of equations obtained in step 3. Then solve the matrix equation.

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Step 5: Substitute the values for a, b, and c back into the standard form of the quadratic function.

d(t) = -16 t2 + 40t + 4

b. Find the coordinates of the vertex and tell what that represents in the real world.

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c. Find the t-intercepts and tell what each represents in the real world. (Use calculator.)

We are looking for whenever d(t) = 0.

0 = -16 t2 + 40t + 4 You may use quadratic formula at this pt.

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t = 2.596 seconds

d. Graph the function.

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e. By looking at your graph and thinking about what it represents, figure out a domain for this

function. Why would your model not give reasonable answers for d when the value of t is

- below the domain? - above the domain?

Domain : [0,2.596] no need for negative time & nothing past 2.596 because the ball has hit the ground.

Below the domain---before the ball is kicked.

Above the domain—after the ball hits the ground.

f. What influences in the real world might make your model slightly inaccurate within the

domain?

Wind, human error in bobbling the ball, etc.

g. From your graph and your calculations, tell what the range of your function is.

Range : [0, 29]

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The vertex is (1.25, 29). The maximum height of 29 ft. occurs at 1.25 seconds after the ball is punted.

d(1.25) = -16 (1.25)2 + 40(1.25) + 4

d(1.25) = 29

The ball hits the ground 2.596 seconds after it is kicked.

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