In the equation above, is a positive constant. When the graph of is drawn in the -plane, the graph has exactly 2 distinct -intercepts. Which of the following could be the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 84 | Free SAT Prep | Aniko

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Question 84·Hard·Nonlinear Functions

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f(x)= (x - 4)(x + 1)^2 (x - k)^2

In the equation above, $k$ is a positive constant. When the graph of $y = f(x)$ is drawn in the $xy$ -plane, the graph has exactly 2 distinct $x$ -intercepts. Which of the following could be the value of $k$ ?

For polynomial x-intercept questions where the function is already factored, immediately set each factor equal to zero to find the roots. Pay close attention to words like “distinct” and to any conditions on parameters (such as $k>0$ ), because these tell you which roots can or cannot coincide. Then enforce the intercept condition (for example, “exactly 2 distinct x-intercepts”) by deciding which roots must be equal or excluded, solve for the parameter, and finally check that your value appears in the answer choices.

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Use the factored form to find zeros

Remember that x-intercepts occur where $y=0$ . For a factored polynomial, this happens when any factor equals 0. Write down the values of $x$ that make $(x - 4)$ , $(x + 1)^2$ , and $(x - k)^2$ equal to 0.

Think about the sign of $k$

You know $k$ is positive. How does that compare to the value $-1$ ? Can $k$ ever be equal to $-1$ if $k>0$ ?

Connect “distinct x-intercepts” to the roots you found

You will get three expressions for possible x-intercepts: one negative, one fixed positive, and one involving $k$ . For the graph to have exactly 2 distinct x-intercepts, what must be true about these three numbers? Which equality between them is actually possible when $k>0$ ?

Desmos Guide

Set up the general function with a parameter

In Desmos, type:
y = (x - 4)(x + 1)^2 (x - a)^2
Desmos will create a slider for a, which represents the value of $k$ .

Test each candidate value of k

Move the slider so that $a=1$ , then $a=3$ , then $a=4$ , and then $a=5$ . For each setting, look at the graph and count how many distinct x-values where the graph touches or crosses the x-axis.

Identify the correct value from the graph

Among $a=1,3,4,5$ , find the value of $a$ for which the graph of $y=(x - 4)(x + 1)^2 (x - a)^2$ has exactly 2 distinct x-intercepts. That value of $a$ is the correct value of $k$ .

Step-by-step Explanation

Find the x-intercepts from the factored form

The x-intercepts of $y=f(x)$ are the $x$ -values where $f(x)=0$ .

Given

f(x) = (x - 4)(x + 1)^2 (x - k)^2,

set each factor equal to 0:

$x - 4 = 0 \Rightarrow x = 4$
$(x + 1)^2 = 0 \Rightarrow x + 1 = 0 \Rightarrow x = -1$
$(x - k)^2 = 0 \Rightarrow x - k = 0 \Rightarrow x = k$

So the possible x-intercepts are $x=4$ , $x=-1$ , and $x=k$ (each counted once as a distinct x-value, even if the factor has an exponent).

Use the condition that $k$ is positive

We are told $k$ is a positive constant, so $k>0$ .

Among the three possible intercepts $x=4$ , $x=-1$ , and $x=k$ :

$x=-1$ is negative and fixed; it cannot equal a positive number.
$x=4$ and $x=k$ are both nonnegative when $k>0$ .

Because $k>0$ , $k$ cannot equal $-1$ , so $x=-1$ will always be one distinct x-intercept.

Translate “exactly 2 distinct x-intercepts” into a condition on the roots

Right now, the three potential roots (zeros) are:

$x=-1$
$x=4$
$x=k$

If all three are different numbers, then the graph will have 3 distinct x-intercepts. The problem says there are exactly 2 distinct x-intercepts, which means two of these three values must be equal, so that they represent the same point on the $x$ -axis.

The possible equalities are:

$-1 = 4$ (impossible),
$-1 = k$ (impossible because $k>0$ ),
$4 = k$ (this one is possible).

So the only way to reduce the number of distinct x-intercepts from 3 to 2 is to have $4$ and $k$ be the same number.

Solve for $k$ and match the answer choice

From the previous step, we need $4 = k$ .

So $k = 4$ .

This matches answer choice C) $4$ , which is the correct answer.

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