Question

The function h(t)=-5t^2+20t+60 models the height in meters of a ball t seconds after it’s thrown . Which describe the intercepts and vertex of this function

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Santino

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To find the intercepts of the function h(t) = -5t^2 + 20t + 60, we need to set it equal to zero and solve for t.

1. Set h(t) = 0:

-5t^2 + 20t + 60 = 0

2. Solve the quadratic equation using the quadratic formula:

t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a = -5, b = 20, and c = 60. Plugging these values into the quadratic formula:

t = \frac{-20 \pm \sqrt{20^2 - 4(-5)(60)}}{2(-5)}

3. Simplify the equation:

t = \frac{-20 \pm \sqrt{400 + 1200}}{-10}

t = \frac{-20 \pm \sqrt{1600}}{-10}

t = \frac{-20 \pm 40}{-10}

4. Solve for t:

Case 1:

t = \frac{-20 + 40}{-10}

t = -2

Case 2:

t = \frac{-20 - 40}{-10}

t = 6

Therefore, the intercepts of the function h(t) are t = -2 and t = 6. This means that the graph of the function crosses the t-axis at t = -2 and t = 6.

To find the vertex of the function, we can use the formula for the x-coordinate of the vertex, which is given by:

x = -\frac{b}{2a}

In this case, a = -5 and b = 20. Plugging these values into the formula:

x = -\frac{20}{2(-5)}

x = -\frac{20}{-10}

x = 2

To find the y-coordinate of the vertex, we substitute the x-coordinate back into the function:

h(2) = -5(2)^2 + 20(2) + 60

h(2) = -20 + 40 + 60

h(2) = 80

Therefore, the vertex of the function h(t) is (2, 80).

Answer:

The intercepts of the function h(t) = -5t^2 + 20t + 60 are t = -2 and t = 6.

The vertex of the function h(t) is (2, 80).

1. Set h(t) = 0:

-5t^2 + 20t + 60 = 0

2. Solve the quadratic equation using the quadratic formula:

t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a = -5, b = 20, and c = 60. Plugging these values into the quadratic formula:

t = \frac{-20 \pm \sqrt{20^2 - 4(-5)(60)}}{2(-5)}

3. Simplify the equation:

t = \frac{-20 \pm \sqrt{400 + 1200}}{-10}

t = \frac{-20 \pm \sqrt{1600}}{-10}

t = \frac{-20 \pm 40}{-10}

4. Solve for t:

Case 1:

t = \frac{-20 + 40}{-10}

t = -2

Case 2:

t = \frac{-20 - 40}{-10}

t = 6

Therefore, the intercepts of the function h(t) are t = -2 and t = 6. This means that the graph of the function crosses the t-axis at t = -2 and t = 6.

To find the vertex of the function, we can use the formula for the x-coordinate of the vertex, which is given by:

x = -\frac{b}{2a}

In this case, a = -5 and b = 20. Plugging these values into the formula:

x = -\frac{20}{2(-5)}

x = -\frac{20}{-10}

x = 2

To find the y-coordinate of the vertex, we substitute the x-coordinate back into the function:

h(2) = -5(2)^2 + 20(2) + 60

h(2) = -20 + 40 + 60

h(2) = 80

Therefore, the vertex of the function h(t) is (2, 80).

Answer:

The intercepts of the function h(t) = -5t^2 + 20t + 60 are t = -2 and t = 6.

The vertex of the function h(t) is (2, 80).

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