The height of a thrown ball is a quadratic function of the time it has been in the air. The graph of the quadratic function is the parabolic path of the ball. The vertex of the graph is (1, 20) and the path of the ball includes the point (0,4). What is an expression that defines this function? Write the quadratic function in vertex form.

Let's look at the given data: h(t)= -16 t2 +80t

Question #1: How high does the ball go? (maximum height)

Use the vertex form for a parabola: y= a(X-h)2 +K .

(If "a" is positive, the parabola opens upward, and if "a" is negative, it opens downward. Also, the vertex of the

parabola is (h,K).) You will want to complete the square on h(t)= -16 t2 +80t .

Completing the Square

1. Factor out -16 from both terms: -16( t2 - 5t)

2. Take half of the middle term constant: -5/2 and square it: (-5/2)2 = (25)/4

3. You make a third term with it by adding it to the problem, but in a way, you also have to subtract it, so that you

are not breaking the math rules.

5/2= 2.5

Simplify

h K

-16( t2-5t + 25/4) +(-25/4 )(-16)= -16 ( t- 5/2)2 + 100 Vertex Form ( 2.5, 100)

Therefore, at 2.5 seconds up, the ball is at its maximum height of 100 feet.

Question 2: How many seconds does it take for the ball to hit the ground? You're looking for an intercept at the

ground, where the height is zero feet.

1. Just factor out "-16t " the original equation: h(t) = -16 t2 +80t = -16t (t-5)

h(t)=0 -16t =0 or t-5 =0

t=0 or t=5 seconds V(5,0)

Don't use t=0 because that was before the ball started in motion. t=5, means that it was traveling 5 seconds when

it hit the ground.

You may check work by using a graphing calculator; just change Y max to about 100 feet. Use Window function. Let X min be 0 and X max 10, y min =0 , and y max =100. The X axis represents time (t), and the y- axis represents height of ball. Next just change the equation to the following: y= -16X2+ 80X. Enter it into the y function mode and press Graph. You should be able to see the Maximum point (vertex) of a parabola opening downward. Then press 2nd Graph, and you can see the table and check the values to see if they are the same

f(x) = a(x - 1) + 20

Step-by-step explanation:

Vertex form is written in this format: f(x) = a(x - h) + k. The vertex point is written in the form (h,k). To write the vertex in the equation, fill in the digits where they belong.

Hope it helps!

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step-by-step explanation: