SAT Two-Variable Data Question 30 | Free SAT Prep | Aniko

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Question 30·Hard·Two-Variable Data: Models and Scatterplots

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The table shows values of $x$ and $y$ for a set of measurements. The $y$ -values are rounded to the nearest whole number.

$x$	$y$
$1$	$510$
$4$	$313$
$7$	$193$
$10$	$118$

An exponential model of the form $y=a\,b^x$ is proposed, where $a>0$ and $0<b<1$ .

Which choice gives a model that is most consistent with the data?

For $y=a\,b^x$ , use two points to form a ratio $\frac{y_2}{y_1}=b^{\Delta x}$ , then take the appropriate root to estimate $b$ . After that, substitute any point into $y=a\,b^x$ to find $a$ , and verify the choice by checking how well it matches another data point (allowing for rounding).

Hints

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Use ratios for exponential models

With an exponential model, compare two points using a ratio like $\frac{y(4)}{y(1)}$ .

Match the ratio to the change in $x$

From $x=1$ to $x=4$ , the change in $x$ is 3, so the ratio of the $y$ -values corresponds to $b^3$ .

Find $a$ after you have $b$

Once you have an estimate for $b$ , plug one data point (like $x=1$ ) into $y=a\,b^x$ to solve for $a$ .

Desmos Guide

Enter the data points

In a table, enter the points $(1,510)$ , $(4,313)$ , $(7,193)$ , and $(10,118)$ .

Graph the candidate models

In separate lines, graph each option:

$y=600(0.85)^x$
$y=600(0.84)^x$
$y=510(0.85)^x$
$y=600(0.85)^{x/3}$

Compare closeness to the points

Check which curve stays closest to all four plotted points (smallest vertical gaps overall), remembering the $y$ -values are rounded.

Step-by-step Explanation

Use a ratio over a change in $x$

In $y=a\,b^x$ , if $x$ increases by 3, then $y$ is multiplied by $b^3$ .

Using $x=1$ and $x=4$ from the table:

\frac{y(4)}{y(1)}\approx \frac{313}{510}\approx 0.614

So $b^3\approx 0.614$ .

Estimate $b$ by taking a cube root

Take the cube root of both sides:

b\approx \sqrt[3]{0.614}\approx 0.85

(For example, $(0.85)^3\approx 0.614$ .)

Use a data point to estimate $a$

Use the row where $x=1$ and $y\approx510$ :

510\approx a(0.85)^1

a\approx \frac{510}{0.85}=600.

Write the model

Substituting the estimated parameters gives the model

y=600(0.85)^x.

Check with another point (rounding is expected)

Check $x=7$ :

600(0.85)^7\approx 600(0.3206)\approx 192.4\approx 193,

which matches the table after rounding. Therefore, the most consistent choice is $y=600(0.85)^x$ .

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