Question 30·SAT Desmos Regression Practice·Algebra

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A linear function $f$ satisfies the conditions

$2f(1)+f(7)=20$
$f(4)=6$

What is the value of $f(10)$ ?

Your answer

A linear function changes by the same amount each time its input changes by the same amount. If you’re given a combination of function values rather than the rule, write $f(x)=ax+b$ and use a matched-list regression in Desmos to fit both conditions at once. Don’t treat an output at a nonzero input as the starting value.

Hints

Hint 1
Write the linear function as $f(x)=ax+b$ . The constants $a$ and $b$ must be the same in both conditions. What do $f(1)$ , $f(7)$ , and $f(4)$ become?
Hint 2
A matched-list regression pairs entries by position. Put the two expressions from your conditions in one list and their stated values in the other, so Desmos fits one pair of constants.
Hint 3
The fitted constants describe the function, but the question asks for an output at a new input. After the fit, define $f$ using those constants and evaluate the requested input.

Step-by-step

Approach 1: Fit the linear function in Desmos
Step 1
Turn both conditions into equations
For a linear function, $f(x)=ax+b$ : $a$ is the change in output when the input rises by $1$ , and $b=f(0)$ . Replace each function value with its expression:
$2(a+b)+(7a+b)=20,\qquad 4a+b=6.$
The same $a$ and $b$ appear in both equations. Don’t set $b=6$ ; $6$ is $f(4)$ , not $f(0)$ .
Step 2
Fit both conditions together
Type $[2(a+b)+(7a+b),4a+b]\sim[20,6]$ . The regression pairs the first expressions and values, then the second, while fitting one shared pair of constants. Desmos reports $a\approx-0.6667$ and $b\approx8.6667$ under PARAMETERS.
Step 3
Evaluate the requested output
Below the regression, type $f(x)=ax+b$ , then $f(10)$ . Desmos uses the fitted constants and displays $f(10)=2$ . So the output at input $10$ is $2$ . Grid in 2.

Approach 2: Use equal spacing instead
Step 1
Find the midpoint relationship
The inputs $1$ , $4$ , and $7$ are equally spaced. Equal input steps mean equal output steps for a linear function, so $f(4)$ is the midpoint of the other two outputs:
$f(1)+f(7)=2f(4)=12.$
Step 2
Match the coefficient of the first output
Double the midpoint equation so its $f(1)$ term will match the given condition:
$2f(1)+2f(7)=24.$
Step 3
Cancel the matching terms
Subtract the given condition from the doubled equation; the $2f(1)$ terms cancel:
$\begin{aligned} (2f(1)+2f(7))-(2f(1)+f(7))&=24-20\\[1.4em] f(7)&=4. \end{aligned}$
Step 4
Continue the equal output change
The inputs $4$ , $7$ , and $10$ also step by $3$ , so add the same output change from $4$ to $7$ once more:
$\begin{aligned} f(10)&=f(7)+(f(7)-f(4))\\[1.4em] f(10)&=4+(4-6)\\[1.4em] f(10)&=2. \end{aligned}$
The output at input $10$ is $2$ . Grid in 2.

Question 30·SAT Desmos Regression Practice·Algebra

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Step-by-step

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