The graph of the function , defined by is shown in the -plane. If the function (not shown) is defined by , what is one possible value of such that ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 3 | Free SAT Prep | Aniko

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Question 3·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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The graph of the function $f$ , defined by

f(x) = -\tfrac12 (x-4)^2 + 10,

is shown in the $xy$ -plane.

If the function $g$ (not shown) is defined by $g(x) = -x + 10$ , what is one possible value of $a$ such that $f(a) = g(a)$ ?

When a question asks for a value where two functions are equal, set their expressions equal and solve the resulting equation. For a quadratic and a line, this will give a quadratic equation: simplify carefully (clear constants and fractions first), move everything to one side to get it in standard form, and then factor or use the quadratic formula. Finally, compare your solutions to the answer choices—eliminate any that are not listed and pick the matching value. If you are unsure with algebra under time pressure, you can also quickly test each choice by plugging it into both functions and seeing if the outputs match.

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Connect the condition to an equation

The condition $f(a)=g(a)$ means the $y$ -values of the two functions are the same at $x=a$ . How can you write this using the given formulas for $f$ and $g$ ?

Clear constants and fractions

Once you have written $-\tfrac12(a-4)^2 + 10 = -a + 10$ , try subtracting 10 from both sides, then multiply by $-2$ to get rid of the fraction.

Recognize the quadratic

After simplifying, you should get an equation involving $a^2$ , $a$ , and a constant, all on one side equal to 0. Think about how to factor this trinomial or use the quadratic formula.

Use the solutions with the choices

You will get two values for $a$ from the quadratic. Check which of these values actually appears among the answer choices.

Desmos Guide

Enter the two functions

In Desmos, type y = -0.5*(x-4)^2 + 10 on one line and y = -x + 10 on another line so both graphs (the parabola and the line) are shown together.

Find the intersection points

Click or tap where the line and the parabola cross. Desmos will display the coordinates of each intersection point; note the $x$ -values of these points.

Match with the answer choices

Compare the $x$ -values of the intersection points you found to the options 10, 6, 0, and 2, and choose the option that matches one of those $x$ -values.

Step-by-step Explanation

Set the two functions equal

We want values of $a$ where $f(a)=g(a)$ .

Write the equation:

-\tfrac12 (a-4)^2 + 10 = -a + 10.

This equation represents the $x$ -coordinates where the parabola $f$ and the line $g$ intersect.

Simplify the equation

First, subtract 10 from both sides to remove the constant terms:

-\tfrac12 (a-4)^2 = -a.

Now multiply both sides by $-2$ to clear the fraction and negative sign:

(a-4)^2 = 2a.

Next, expand the square on the left:

(a-4)^2 = a^2 - 8a + 16,

so the equation becomes

a^2 - 8a + 16 = 2a.

Get a standard quadratic equation

Move all terms to one side so the equation is equal to 0:

Starting from

a^2 - 8a + 16 = 2a,

subtract $2a$ from both sides:

a^2 - 10a + 16 = 0.

This is a quadratic equation in $a$ that we can solve by factoring or by using the quadratic formula.

Solve the quadratic and match with the choices

Factor the quadratic:

a^2 - 10a + 16 = (a-2)(a-8) = 0.

Set each factor equal to 0:

$a - 2 = 0$ gives $a = 2$ .
$a - 8 = 0$ gives $a = 8$ .

Both values make $f(a)=g(a)$ , but the question asks for one possible value and gives the choices 10, 6, 0, and 2. The only matching value is 2, so the correct answer choice is D) 2.

Strategy

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation