The function is defined above. Which table of values represents ?

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SAT Nonlinear Functions Question 143 | Free SAT Prep | Aniko

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Question 143·Medium·Nonlinear Functions

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k(x) = (x - 3)(x + 2)(x - 1)

The function $k$ is defined above. Which table of values represents $y = k(x) + 2$ ?

x	y
-2	-2
1	1
3	2

x	y
-2	2
1	2
3	2

x	y
-2	0
1	0
3	0

x	y
-2	2
1	3
3	4

For questions that ask for a table of values of a transformed function like $y = k(x) + c$ , focus only on the specific $x$ -values shown in the tables. Quickly evaluate the original function $k(x)$ at each listed $x$ , then apply the transformation (here, adding 2 to each result). Finally, scan the answer choices and pick the table whose $y$ -values match your transformed values; using the structure of factors to spot zeros can make the evaluations much faster.

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Focus on the definition of the new function

You are given $k(x)$ , but the question asks about $y = k(x) + 2$ . Think about what you must do to $k(x)$ to get the new $y$ -values.

Evaluate $k(x)$ at specific points

Look only at the $x$ -values that appear in the tables: $-2$ , $1$ , and $3$ . What is $k(-2)$ , $k(1)$ , and $k(3)$ ?

Use the structure of the factors

Notice that $k(x)$ is a product of $(x-3)$ , $(x+2)$ , and $(x-1)$ . What happens to this product if $x$ makes any one of these factors equal to 0?

Translate $k(x)$ into $k(x)+2$

Once you know $k(-2)$ , $k(1)$ , and $k(3)$ , remember that the actual function is $y = k(x) + 2$ . How does that change each $y$ -value?

Desmos Guide

Enter the function for $y$

In Desmos, type y = (x - 3)(x + 2)(x - 1) + 2 to represent $y = k(x) + 2$ .

Make a table of values

Click the gear icon next to the equation and choose "Add table" (or use the table icon). In the $x$ -column, enter $-2$ , $1$ , and $3$ .

Read off the $y$ -values

Look at the $y$ -values that Desmos gives for $x = -2$ , $1$ , and $3$ . Then choose the answer choice whose table has exactly those same $y$ -values for those $x$ -values.

Step-by-step Explanation

Understand what $y = k(x) + 2$ means

The function $k(x)$ is given by

k(x) = (x - 3)(x + 2)(x - 1)

The new function is $y = k(x) + 2$ . This means: for any $x$ -value, you first compute $k(x)$ , then you add 2 to that result to get $y$ .

So for each $x$ in the tables, you need to find $k(x)$ and then do $k(x) + 2$ .

Compute $k(x)$ at the given $x$ -values

The tables only use $x = -2, 1, 3$ . Plug each into $k(x)$ :

For $x = -2$ :

k(-2) = (-2 - 3)(-2 + 2)(-2 - 1) = (-5)(0)(-3) = 0

For $x = 1$ :

k(1) = (1 - 3)(1 + 2)(1 - 1) = (-2)(3)(0) = 0

For $x = 3$ :

k(3) = (3 - 3)(3 + 2)(3 - 1) = (0)(5)(2) = 0

So at all three $x$ -values, $k(x) = 0$ .

Add 2 to get $y = k(x) + 2$ and choose the matching table

Now compute $y = k(x) + 2$ for each $x$ :

At $x = -2$ : $y = 0 + 2 = 2$
At $x = 1$ : $y = 0 + 2 = 2$
At $x = 3$ : $y = 0 + 2 = 2$

So the correct table must have $y = 2$ for $x = -2, 1, 3$ . That matches choice B, which shows $(-2,2)$ , $(1,2)$ , and $(3,2)$ .

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