The table shows several values of and their corresponding values of . | | | |----|----| | | | | | | | | | | | | Which of the following functions best models the relationship between and ?

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SAT Nonlinear Functions Question 1 | Free SAT Prep | Aniko

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Question 1·Easy·Nonlinear Functions

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The table shows several values of $x$ and their corresponding values of $y$ .

$x$	$y$
$-1$	$4$
$0$	$2$
$1$	$1$
$2$	$0.5$

Which of the following functions best models the relationship between $x$ and $y$ ?

When a question gives a table of values and asks which function fits, first decide if the pattern is additive (constant difference) or multiplicative (constant ratio). Compute a few successive differences and ratios between y-values: a constant difference suggests a linear model, while a constant ratio suggests an exponential model. Once you identify the type, use an easy point like $x = 0$ or $x = 1$ to find specific parameters, then quickly plug that information into the answer choices or test one or two points in each remaining choice to see which matches all of the table’s values.

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Check how y changes when x increases by 1

Look at the sequence of y-values: 4, 2, 1, 0.5. Are you adding/subtracting the same amount each time, or multiplying by the same factor?

Decide what type of function fits that pattern

A constant difference suggests a linear function; a constant ratio suggests an exponential function. Which situation do you see in this table?

Use the value at x = 0

Once you know the pattern (the base of the exponential), use the point where $x = 0$ to find the leading constant in front of the expression. Then look for the answer choice in that form.

Desmos Guide

Enter the data points

In Desmos, type the four points from the table as $( -1, 4 )$ , $(0, 2)$ , $(1, 1)$ , and $(2, 0.5)$ on separate lines so you can see the plotted points.

Graph each answer choice

On new lines, enter each function from the choices: y = 2*2^x, y = x^2 - 2, y = (x - 1)^3, and y = 2*(1/2)^x. Make sure all four graphs are turned on.

Compare the graphs to the points

Zoom as needed and observe which graph passes exactly through all four plotted points. The function whose curve goes through each of the points $(-1, 4)$ , $(0, 2)$ , $(1, 1)$ , and $(2, 0.5)$ is the correct model.

Step-by-step Explanation

Look for a pattern in how y changes

Check whether the change in $y$ is by a constant amount (linear/quadratic) or by a constant factor (exponential).

Differences: $4 \to 2$ (change $-2$ ), $2 \to 1$ (change $-1$ ), $1 \to 0.5$ (change $-0.5$ ). These are not constant.
Ratios: $2/4 = 1/2$ , $1/2 = 1/2$ , $0.5/1 = 1/2$ . The ratio is constant: each step in $x$ multiplies $y$ by $1/2$ .

A constant ratio means the relationship is exponential, not linear or quadratic or cubic.

Write the general exponential form and use the table

For an exponential pattern with a constant ratio, we can use the model

y = a \cdot b^x

Here, the constant ratio $b$ is $1/2$ , because each time $x$ increases by 1, $y$ is multiplied by $1/2$ . So the model has the form

y = a \cdot \left(\frac{1}{2}\right)^x

To find $a$ , use the point where $x = 0$ . From the table, when $x = 0$ , $y = 2$ .

Determine the full equation and match it to a choice

Substitute $x = 0$ and $y = 2$ into $y = a \cdot (\tfrac{1}{2})^x$ :

When $x = 0$ , $(\tfrac{1}{2})^0 = 1$ , so $y = a \cdot 1 = a$ .
The table says $y = 2$ at $x = 0$ , so $a = 2$ .

Thus the model is

y = 2\left(\frac{1}{2}\right)^x

Comparing with the answer choices, this corresponds to choice D.

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