Question

during a softball game kay hit a fly ball the function f(x) = -16t^2 + 64t + 4 describes the height of the softball in feet. make a table of values for the function and then graph it.

Answer 1

Hello!

The graph is attached.

To graph a parabola we need to know the following:

- If the parabola is open upwards or downwards

- They axis intercepts (if they exist)

- The vertex position (point)

We are given the function:

`f(t)=-16t^(2)+64t+4`

Where,

`a=-16\\b=64\\c=4`

For this case, the coefficient of the quadratic term (a) is negative, it means that the parabola opens downwards.

Finding the axis interception points:

Making the function equal to 0, we can find the x-axis (t) intercepts, but since the equation is a function of the time, we will only consider the positive values, so:

`f(t)=-16t^(2)+64t+4\\0=-16t^(2)+64t+4\\-16t^(2)+64t+4=0`

Using the quadratic equation:

`\frac{-b+-\sqrt{b^(2)-4ac } }{2a}=\frac{-64+-\sqrt{64^(2)-4*-16*4} }{2*-16}\\\\\frac{-64+-\sqrt{64^(2)-4*-16*4} }{2*-16}=(-64+-√(4096+256) )/(-32)\\\\(-64+-√(4096+256) )/(-32)=(-64+-(65.96) )/(-32)\\\\t1=(-64+(65.96) )/(-32)=-0.06\\\\t2=(-64-(65.96) )/(-32)=4.0615`

So, at t=4.0615 the height of the softball will be 0.

Since we will work only with positive values of "x", since we are working with a function of time:

Let's start from "t" equals to 0 to "t" equals to 4.0615.

So, evaluating we have:

`f(0)=-16(0)^(2)+64(0)+4=4\\\\f(1)=-16(1)^(2)+64(1)+4=52\\\\f(2)=-16(2)^(2)+64(2)+4=68\\\\f(3)=-16(3)^(2)+64(3)+4=52\\\\f(4.061)=-16(4.0615)^(2)+64(4.0615)+4=0.0034=0`

Finally, we can conclude that:

- The softball reach its maximum height at t equals to 2. (68 feet)

- The softball hits the ground at t equals to 4.0615 (0 feet)

- At t equals to 0, the height of the softball is equal to 4 feet.

See the attached image for the graphic.

Have a nice day!