Question 1·Easy·Advanced Math

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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 $y$ ?

Equally spaced inputs invite a check for an exponential model: equal input steps multiply the output by the same factor. Read the output at $x=0$ for the number in front of the power, then find the factor. A Desmos exponential regression can confirm both. Don’t choose a base greater than $1$ for outputs that keep shrinking.

Hints

Hint 1
A ratio compares one output with the one before it. When $x$ rises by $1$ , look at $4$ , $2$ , $1$ , and $0.5$ . Are you repeatedly adding the same amount or multiplying by the same number?
Hint 2
In an exponential model $y=ab^x$ , $a$ is the output at $x=0$ because $b^0=1$ . Which table entry gives you that starting value?

Step-by-step

Fit an exponential model to the table
Step 1
Enter the paired values
Put each $x$ -value beside its corresponding $y$ -value in a Desmos table. Desmos plots four falling points. Because $x$ rises by $1$ from row to row, you can check for a common factor, meaning the same multiplier at every step.
Step 2
Find the starting value and factor
Type $y_1\sim ab^{x_1}$ below the table. The tilde tells Desmos to fit the exponential model $y=ab^x$ . Under PARAMETERS, it shows $a=2$ and $b=0.5$ . Here $a$ is the output at $x=0$ , and $b$ is the multiplier for each $1$ -unit increase in $x$ . That factor of $0.5$ matches the table: each output is half the one before it.
Step 3
Match the function to the table
Put Desmos’s values into $y=ab^x$ : $y=2(0.5)^x=2\left(\frac12\right)^x$ . The base $\frac12$ halves the output each step, so this function matches the table. Choice D.

Question 1·Easy·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this