find the vertex for the preceding algebraically:

h(t)=-16t^2+8t+8000

& then use the t- intercepts and the vertex to define a reasonable window for a graphing calculator. please help!! & thank you so much

h(t)=-8(2t^2-t-1000)

= -8(2t -45 )(t+22) about...

so t= about 22.5 and -22
I would graph it fom -20 to 20

To find the vertex of the parabolic function h(t) = -16t^2 + 8t + 8000 algebraically, you can follow these steps:

Step 1: Recall that the vertex of a parabola in the form y = ax^2 + bx + c can be found using the formula x = -b / (2a).

In the given function h(t) = -16t^2 + 8t + 8000, a = -16 and b = 8. Substituting these values into the formula:

t = -8 / (2*-16) = -8 / -32 = 1/4 = 0.25

So, the x-coordinate of the vertex is 0.25.

Step 2: Find the y-coordinate of the vertex by substituting this x-coordinate back into the function h(t):

h(0.25) = -16(0.25)^2 + 8(0.25) + 8000
= -16 * 0.0625 + 2 + 8000
= -1 + 2 + 8000
= 8001

Thus, the y-coordinate of the vertex is 8001.

Therefore, the vertex of the function h(t) = -16t^2 + 8t + 8000 is (0.25, 8001).

To define a reasonable window for the graphing calculator using the t-intercepts and the vertex, you can consider the following steps:

Step 1: Find the t-intercepts by setting h(t) = 0 and solving for t.

-16t^2 + 8t + 8000 = 0

You can either factor or use the quadratic formula to solve this equation. It turns out that the equation does not factor easily, so using the quadratic formula is more suitable.

Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a):

t = (-8 ± √(8^2 - 4*(-16)*(8000))) / (2*(-16))
t = (-8 ± √(64 + 512000)) / (-32)
t = (-8 ± √512064) / (-32)

t ≈ (-8 + 715.027) / (-32) ≈ -22.68 (approximately)
t ≈ (-8 - 715.027) / (-32) ≈ 22.50 (approximately)

So, the t-intercepts are approximately -22.68 and 22.50.

Step 2: Determine a reasonable range for the x-axis on the graphing calculator.
Based on the t-intercepts, it would be reasonable to set the x-axis ranging from slightly left of -22.68 to slightly right of 22.50.

For example, you could set the x-axis from -25 to 25.

Step 3: Determine a reasonable range for the y-axis on the graphing calculator.
Consider the y-coordinate of the vertex, which is 8001, and the shape of the parabola. It would be reasonable to set the y-axis slightly above and below 8001 to capture the full shape of the graph.

For example, you could set the y-axis from 7900 to 8100.

By setting the x-axis from -25 to 25 and the y-axis from 7900 to 8100, you will have a reasonable window to graph h(t) on a calculator, which includes the t-intercepts and the vertex.