An owl swoops down from its perch in a tree at 20 feet to catch field mice below. It follows the function h=−6t2−24t+20 . Plot points to graph the equation. Which ordered pair represents the vertex?(1 point)

Question

An owl swoops down from its perch in a tree at 20 feet to catch field mice below. It follows the function h=−6t2−24t+20 . Plot points to graph the equation. Which ordered pair represents the vertex?(1 point)

Responses

(1,0)
left parenthesis 1 comma 0 right parenthesis

(44,−2)
left parenthesis 44 comma negative 2 right parenthesis

(−2,44)
left parenthesis negative 2 comma 44 right parenthesis

(−5,0)

Answer 1

To graph the equation h = -6t^2 - 24t + 20, we can start by plotting a few points. However, to find the vertex of the parabola represented by the quadratic function, we don't necessarily need to plot many points because the vertex form of a quadratic function is given by:

h(t) = a(t - h)^2 + k,

where (h, k) is the vertex of the parabola. In the given equation, we can use the formula for the vertex of a parabola in standard form:

h(t) = at^2 + bt + c.

The vertex (h, k) can be found using the formula:

h = -b/(2a),
k = c - (b^2 - 4ac) / (4a).

Given the equation h = -6t^2 - 24t + 20, a = -6, b = -24, and c = 20. We can find the x-coordinate of the vertex h:

h = -(-24)/(2 * -6) = 24/12 = 2.

Now let's find the y-coordinate k:

k = 20 - ((-24)^2 - 4 * -6 * 20) / (4 * -6) = 20 - (576 + 480) / -24 = 20 - 1056 / -24 = 20 + 44 = 64.

So the vertex of the parabola is at the point (2, 64).

Now let's plot a few points around the vertex to sketch the graph. You can choose t values of 0, 1, 2, etc., and calculate the corresponding h values. Here is a sample set of points:

When t = 0: h = -6(0)^2 - 24(0) + 20 = 20. (0, 20)
When t = 1: h = -6(1)^2 - 24(1) + 20 = -6 - 24 + 20 = -10. (1, -10)
When t = 3: h = -6(3)^2 - 24(3) + 20 = -54 - 72 + 20 = -106. (3, -106)

And we already know the vertex is at (2, 64).

Plotting these points and the vertex, we get a parabola opening downward, as the leading coefficient (-6) is negative.

None of the answer choices provided in the question corresponds to the correct vertex, which is (2, 64). There might be a mistake in the question or answer choices if it's expected that one of the given pairs is the correct answer for the vertex of the given quadratic equation.