The scatterplot shows the relationship between two variables, and , for data set A. A line of best fit is shown. Data set B is created from data set A by applying both of the following changes to each data point: - The -coordinate is increased by and then multiplied by . - The -coordinate is decreased by and then multiplied by . Which choice could be an equation of a line of best fit for data set B?

Solution available with step-by-step explanation

SAT Two-Variable Data Question 45 | Free SAT Prep | Aniko

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Question 45·Hard·Two-Variable Data: Models and Scatterplots

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The scatterplot shows the relationship between two variables, $x$ and $y$ , for data set A. A line of best fit is shown.

Data set B is created from data set A by applying both of the following changes to each data point:

The $x$ -coordinate is increased by $2$ and then multiplied by $5$ .
The $y$ -coordinate is decreased by $1$ and then multiplied by $3$ .

Which choice could be an equation of a line of best fit for data set B?

First, extract the line of best fit from the graph using two clear points on the drawn line (this gives you slope and intercept quickly). Then treat the coordinate changes as algebraic substitutions: write the new coordinates in terms of the old ones, solve for the old variable (usually $x$ ) in terms of the new one, and substitute into the original line. Finally, apply any scaling/shifting to $y$ and simplify to a standard form so you can compare directly to the answer choices.

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Read two clear points on the drawn line

Use the two labeled points that lie exactly on the line of best fit to find its slope and intercept.

Turn the word changes into equations

Write the new coordinates as $X=5(x+2)$ and $Y=3(y-1)$ , where $(x,y)$ is from data set A.

Express the old $x$ using the new $X$

Solve $X=5(x+2)$ for $x$ , then substitute that expression into the line for data set A before applying $Y=3(y-1)$ .

Desmos Guide

Compute the slope and intercept from the graphed line

Enter

m=(14-4)/(4-0)
b=4

so the line for data set A is y=m x + b.

Write the transformation using new-variable notation

Use X as the new $x$ -variable. From $X=5(x+2)$ , enter xold = X/5 - 2.

Create an expression for the transformed line

Enter

Y = 3*((m*xold + b) - 1)

Desmos will display $Y$ as a simplified linear expression in $X$ .

Match the displayed line to a choice

Compare the coefficient of $X$ (the slope) and the constant term (the intercept) in Desmos’s simplified expression to the four answer choices.

Step-by-step Explanation

Find an equation for the line of best fit in data set A

From the graph, the line of best fit passes through the labeled points $(0,4)$ and $(4,14)$ .

So the slope is

\frac{14-4}{4-0}=\frac{10}{4}=\frac{5}{2}

Since the line goes through $(0,4)$ , the $y$ -intercept is $4$ . So an equation is

y = \frac{5}{2}x + 4.

Rewrite the coordinate changes as equations

Let $(x,y)$ be a point in data set A, and let $(X,Y)$ be the corresponding point in data set B.

“Increase $x$ by $2$ and then multiply by $5$ ” means $X = 5(x+2)$ .
“Decrease $y$ by $1$ and then multiply by $3$ ” means $Y = 3(y-1)$ .

Substitute to write $Y$ in terms of $X$

Solve $X = 5(x+2)$ for $x$ :

X=5x+10 \quad \Rightarrow \quad x=\frac{X-10}{5}=\frac{X}{5}-2.

Use the line from data set A, $y = \frac{5}{2}x + 4$ , inside $Y = 3(y-1)$ :

\begin{aligned} Y &= 3\left(\left(\frac{5}{2}x+4\right)-1\right) \\ &= 3\left(\frac{5}{2}x+3\right). \end{aligned}

Now substitute $x=\frac{X}{5}-2$ :

\begin{aligned} Y &= 3\left(\frac{5}{2}\left(\frac{X}{5}-2\right)+3\right) \\ &= 3\left(\frac{1}{2}X-5+3\right) \\ &= 3\left(\frac{1}{2}X-2\right). \end{aligned}

Simplify to match an answer choice

Distribute the $3$ :

Y = \frac{3}{2}X - 6.

So an equation of a line of best fit for data set B could be $y = \frac{3}{2}x - 6$ .

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