The function is defined by In the -plane, the graph of is obtained by performing the following transformations on the graph of , in the order given: 1. Reflect the graph across the -axis. 2. Translate the resulting graph units to the right. 3. Translate the resulting graph units down. What is the value of ?

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SAT Nonlinear Functions Question 177 | Free SAT Prep | Aniko

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Question 177·Hard·Nonlinear Functions

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The function $f$ is defined by

f(x) = (x-4)(x+1)(x+5).

In the $xy$ -plane, the graph of $y = h(x)$ is obtained by performing the following transformations on the graph of $y = f(x)$ , in the order given:

Reflect the graph across the $y$ -axis.
Translate the resulting graph $3$ units to the right.
Translate the resulting graph $7$ units down.

What is the value of $h(2)$ ?

Your answer

(Express the answer as an integer)

For transformation questions, convert each geometric move into a precise algebra change: reflection across the $y$ -axis means replace $x$ with $-x$ , a shift right by $a$ means use $x-a$ , and a shift down by $b$ means subtract $b$ from the whole function. Apply these changes in the exact order given to build the new function, simplify if helpful, and then substitute the requested $x$ -value. Double-check signs (especially from reflections and vertical shifts), since most mistakes come from missing or misapplied negatives.

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Turn each transformation into a change in the formula

Think about how each transformation (reflect, shift right, shift down) changes the equation of the function. Ask: does it change $x$ inside the function, or does it add/subtract something outside?

Reflection across the y-axis

For a reflection across the $y$ -axis, in the formula for $f(x)$ , every $x$ becomes $-x$ . Write a new function $g_1(x)$ that equals $f(-x)$ .

Horizontal and vertical shifts

A shift right 3 units means replace $x$ with $x-3$ in the current function. A shift down 7 units means subtract 7 from the entire function value. Apply these in the given order to build $h(x)$ .

Evaluate at x = 2

Once you have an explicit formula for $h(x)$ after all transformations, plug in $x=2$ and simplify carefully, watching the negative signs.

Desmos Guide

Enter the original function

In Desmos, type f(x) = (x - 4)(x + 1)(x + 5) to define the original function.

Define the transformed function step by step (or directly)

Either define each step:

g1(x) = f(-x) (reflection across the y-axis)
g2(x) = g1(x - 3) (shift right 3)
h(x) = g2(x) - 7 (shift down 7)

or define it in one line as h(x) = - (x + 1)(x - 4)(x - 8) - 7.

Evaluate h(2)

In a new line, type h(2) or create a table for h(x) and include $x=2$ . Read off the numeric result Desmos displays; that is the value of $h(2)$ .

Step-by-step Explanation

Apply the reflection across the y-axis

The original function is

f(x) = (x-4)(x+1)(x+5).

Reflecting across the $y$ -axis means we replace $x$ with $-x$ in the formula. So the new function after step 1 is

g_1(x) = f(-x) = (-x-4)(-x+1)(-x+5).

We can factor out $-1$ from each factor:

(-x-4)(-x+1)(-x+5) = - (x+4)(x-1)(x-5),

g_1(x) = - (x+4)(x-1)(x-5).

Translate the reflected graph 3 units to the right

To shift a graph of $y=g(x)$ right 3 units, we replace $x$ with $x-3$ in $g(x)$ .

Here $g_1(x) = - (x+4)(x-1)(x-5)$ , so after shifting right 3 we get

g_2(x) = g_1(x-3).

Substitute $x-3$ for $x$ :

g_2(x) = - \big((x-3)+4\big)\big((x-3)-1\big)\big((x-3)-5\big).

Simplify each factor:

$(x-3)+4 = x+1$
$(x-3)-1 = x-4$
$(x-3)-5 = x-8$

g_2(x) = - (x+1)(x-4)(x-8).

Translate the graph 7 units down

To shift a graph of $y=g(x)$ down 7 units, we subtract 7 from the function value: $y = g(x) - 7$ .

Here $g_2(x) = - (x+1)(x-4)(x-8)$ , so the final function is

h(x) = g_2(x) - 7 = - (x+1)(x-4)(x-8) - 7.

Evaluate h(2) using the final formula

Now substitute $x=2$ into $h(x)$ :

h(2) = - (2+1)(2-4)(2-8) - 7.

Compute each factor:

$2+1 = 3$
$2-4 = -2$
$2-8 = -6$

(2+1)(2-4)(2-8) = 3 \times (-2) \times (-6) = 3 \times 12 = 36,

thus

h(2) = -36 - 7 = -43.

So the value of $h(2)$ is $-43$ .

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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