An architect models the height (in feet) of a parabolic arch above the ground. In a coordinate system, represents the horizontal distance (in feet) from the center of the arch, and represents the height (in feet). The highest point of the arch is feet above the ground at . At a point feet from the center, the height of the arch is feet. Which choice could be an equation for ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 53 | Free SAT Prep | Aniko

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Question 53·Medium·Nonlinear Functions

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An architect models the height $h$ (in feet) of a parabolic arch above the ground. In a coordinate system, $x$ represents the horizontal distance (in feet) from the center of the arch, and $h(x)$ represents the height (in feet).

The highest point of the arch is $6$ feet above the ground at $x=0$ . At a point $2$ feet from the center, the height of the arch is $2$ feet.

Which choice could be an equation for $h(x)$ ?

When a quadratic’s maximum or minimum point is given, start with vertex form $a(x-h)^2+k$ using the vertex $(h,k)$ . Then plug in one additional point to solve for $a$ , and finally compare to the choices (including whether the parabola opens upward or downward).

Hints

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Identify the vertex

The highest point of the parabola is its vertex. Translate the sentence about the maximum height into a vertex $(x,y)$ .

Write a vertex-form model

With vertex $(0,6)$ , you can write $h(x)=a(x-0)^2+6$ for some constant $a$ .

Use the point 2 feet from the center

Substitute $x=2$ and $h(2)=2$ to solve for $a$ .

Desmos Guide

Plot the key points

In Desmos, plot the points $(0,6)$ and $(2,2)$ .

Enter a vertex-form equation with a slider

Type $y=a(x-0)^2+6$ and create a slider for $a$ .

Match the second point

Adjust $a$ until the curve passes through $(2,2)$ while keeping the vertex at $(0,6)$ .

Choose the matching option

Pick the answer choice whose equation matches the value of $a$ that fits both points.

Step-by-step Explanation

Use the vertex information

The arch’s highest point is $6$ feet at $x=0$ , so the vertex is $(0,6)$ . A quadratic with vertex $(0,6)$ can be written as

h(x)=a(x-0)^2+6=ax^2+6.

Substitute the additional point to find $a$

The arch is $2$ feet high at $x=2$ , so $h(2)=2$ .

Substitute into $h(x)=ax^2+6$ :

2=a(2)^2+6=4a+6.

So $4a=-4$ , and therefore $a=-1$ .

Select the matching equation

Substitute $a=-1$ into $h(x)=ax^2+6$ :

h(x)=-x^2+6.

Therefore, the correct choice is $h(x)=-x^2+6$ .

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