A projectile is launched from a platform. The height h(t), in meters, of the projectile t … — SAT

A projectile is launched from a platform. The height $h(t)$ , in meters, of the projectile $t$ seconds after it is launched is modeled by the function $h(t) = -5(t - 2)^2 + 24$ . What is the maximum height, in meters, reached by the projectile?

A
2
24Answer
C
-2
D
4

Answer

The maximum height reached by the projectile is 24 meters.

The given quadratic function is in vertex form,

h(t) = a(t - d)^2 + c

, where the vertex of the parabola is

(d, c)

. For the function

h(t) = -5(t - 2)^2 + 24

, the vertex is at

(2, 24)

. Because the leading coefficient

-5

is negative, the parabola opens downward, which means the vertical coordinate of the vertex represents the maximum value of the function. Therefore, the maximum height reached by the projectile is 24 meters.

Step-by-Step Solution

Identify the form of the quadratic function.

The function

h(t) = -5(t - 2)^2 + 24

is given in vertex form, which is

h(t) = a(t - d)^2 + c

Recognizing the vertex form allows direct identification of the vertex of the parabola.

Determine the vertex of the parabola.

For

h(t) = -5(t - 2)^2 + 24

, the vertex is

(2, 24)

The vertex coordinates

(d, c)

represent the input and output values at the turning point of the parabola.

Determine whether the vertex represents a maximum or a minimum.

Since the coefficient

a = -5

is negative, the parabola opens downward, meaning the vertex is the highest point on the graph.

The sign of the leading coefficient determines the concavity of the parabola.

Extract the maximum value of the function.

The maximum value is the vertical coordinate of the vertex, which is 24.

The height of the projectile is represented by the output of the function, which is maximized at the vertex.

Key Concept

Identifying the maximum value of a quadratic function from its vertex form.

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