The function gives the estimated resale value , in thousands of dollars, of a rare collectible years after it was purchased, where . Which statement best interprets the point on the graph of this function?

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SAT Nonlinear Functions Question 179 | Free SAT Prep | Aniko

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Question 179·Medium·Nonlinear Functions

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The function

V(t) = 200(0.75)^t

gives the estimated resale value $V$ , in thousands of dollars, of a rare collectible $t$ years after it was purchased, where $0 \le t \le 8$ . Which statement best interprets the point $(4, 63)$ on the graph of this function?

For questions about interpreting points on the graph of a function, first identify clearly what the input and output represent (including units) from the function’s description. Then read a point $(a,b)$ as: “When the input is $a$ (with units), the output is $b$ (with units).” Translate that into a plain-language sentence and compare it to the answer choices, watching for traps that treat the point as a rate of change or that swap the roles of the coordinates.

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Identify what each axis represents

In the function $V(t) = 200(0.75)^t$ , what does $t$ stand for, and what quantity does $V$ represent, including its units?

Translate the point into words

A point on the graph is written as $(t, V)$ . For the point $(4, 63)$ , which number is the time in years and which number is the value? Say that in a sentence.

Convert from thousands of dollars to dollars

Remember that $V$ is in thousands of dollars. If $V=63$ , how many actual dollars is that? Then combine this with the time to form a complete statement.

Desmos Guide

Graph or evaluate the function

In Desmos, enter the expression 200*(0.75)^x or the function y = 200*(0.75)^x to represent $V(t) = 200(0.75)^t$ .

Find the function value at 4 years

Either type 200*(0.75)^4 on a separate line, or click on the graph where $x=4$ (you can also type x=4 to draw a vertical line and see where it meets the curve). Note the $y$ -value you see there; that is the value of $V$ when $t=4$ , measured in thousands of dollars.

Interpret the output in context

Convert that $y$ -value from thousands of dollars into dollars, then express it in a sentence like: “___ years after purchase, the collectible’s value is ___ dollars.” Finally, pick the answer choice that matches this sentence.

Step-by-step Explanation

Understand what the variables and units mean

The function is

V(t) = 200(0.75)^t

Here:

$t$ is the number of years after the collectible was purchased.
$V$ is the estimated resale value in thousands of dollars.

So a point $(t, V)$ on the graph tells you: $t$ years after purchase, the collectible’s value is $V$ thousand dollars.

Interpret the coordinates of the point (4, 63)

In the point $(4,63)$ :

The $x$ -coordinate $4$ corresponds to $t=4$ years after purchase.
The $y$ -coordinate $63$ corresponds to $V=63$ .

Because $V$ is in thousands of dollars, $V=63$ means the value is $63,000.

Match the interpretation to the answer choices

From Step 2, $(4,63)$ means:

Four years after purchase, the collectible’s estimated value is $63,000.

Now compare with the answer choices:

This sentence exactly matches choice C: “Four years after purchase, the collectible’s estimated value is $63,000.” So C is the correct answer.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation