The height , in meters, of a drone above the ground seconds after takeoff is modeled by the quadratic function According to the model, what is the maximum height, in meters, the drone reaches during the flight?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 43 | Free SAT Prep | Aniko

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Question 43·Easy·Nonlinear Functions

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The height $h$ , in meters, of a drone above the ground $t$ seconds after takeoff is modeled by the quadratic function

h(t) = -2\,(t - 10)^2 + 180.

According to the model, what is the maximum height, in meters, the drone reaches during the flight?

For quadratic word problems on the SAT, first check whether the function is in vertex form $a(x - h)^2 + k$ . If it is, you can read the maximum or minimum value directly from the vertex: if $a < 0$ , the parabola opens downward and the y-value of the vertex is the maximum; if $a > 0$ , it opens upward and the y-value is the minimum. This lets you avoid expanding or doing long calculations—just identify when the squared term is zero and evaluate the function there.

Hints

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Think about the shape of the graph

Look at the coefficient of $(t - 10)^2$ . Is it positive or negative, and what does that tell you about whether the parabola opens upward or downward?

Use the structure of vertex form

The function is written as $-2(t - 10)^2 + 180$ . In a quadratic of the form $a(t - p)^2 + q$ , what does the point $(p, q)$ represent on the graph?

Focus on the squared expression

When is $(t - 10)^2$ as small as possible, and what value does it take then? What happens to $h(t)$ at that moment?

Evaluate at the key time

Once you know the value of $t$ that gives the maximum height, plug that value into $h(t)$ and simplify to find the height.

Desmos Guide

Enter the function

In Desmos, type h(t) = -2*(t - 10)^2 + 180 (or use x instead of t if you prefer: y = -2*(x - 10)^2 + 180). This will graph the height of the drone over time.

View the vertex of the parabola

Look at the highest point on the graph of the parabola. Tap or click on that top point (the vertex) to show its coordinates. The x-coordinate is the time when the drone reaches its maximum height, and the y-coordinate is the maximum height itself.

Step-by-step Explanation

Recognize the form of the quadratic

The function is $h(t) = -2(t - 10)^2 + 180$ . This is vertex form, which looks like $a(t - p)^2 + q$ , where the vertex of the parabola is at $(p, q)$ .

Since the coefficient of the squared term is negative ( $-2$ ), the parabola opens downward, so its vertex represents a maximum point, not a minimum.

Identify when the squared term is smallest

The expression $(t - 10)^2$ is a square, so it is always greater than or equal to $0$ for any real value of $t$ .

It is smallest (actually, zero) when $t = 10$ , because then $t - 10 = 0$ and $(t - 10)^2 = 0$ .

At this time $t = 10$ , the height $h(t)$ will be as large as possible, because we are adding $-2$ times the squared term to $180$ . Making the squared term zero gives the largest possible value for $h(t)$ .

Find the height at the maximum point

Now substitute $t = 10$ into the function to find the height at this time:

h(10) = -2(10 - 10)^2 + 180

$10 - 10 = 0$ , so $(10 - 10)^2 = 0$ , and then

h(10) = -2 \cdot 0 + 180 = 180

So, according to the model, the maximum height the drone reaches is 180 meters, which corresponds to choice B).

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation