Question 7·SAT Desmos Regression Practice·Advanced Math

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A video game awards a bonus score $f(x)$ based on the number of tokens $x$ collected. The bonus is modeled by a quadratic function.

$x$	$f(x)$
0	2
1	5
2	4

Which choice is an equation for $f(x)$ ?

Three input-output pairs and a stated quadratic model point to the form $f(x)=ax^2+bx+c$ . Put the pairs in a Desmos table and run a quadratic regression to find the coefficients, then match each coefficient to its term. Don't treat one change in score as the coefficient of $x$ : the squared term affects that change too.

Hints

Hint 1
Each table row is an input-output pair: the token count is the input, and the bonus score is the output. Put them in Desmos's $x_1$ and $y_1$ columns in matching rows. Which column holds the token counts?
Hint 2
A quadratic regression finds the numbers multiplying $x^2$ and $x$ , plus the constant. Type $y_1\sim ax_1^2+bx_1+c$ under your table; the tilde tells Desmos to fit those numbers to the paired rows.
Hint 3
All the options share one constant term, so checking the score at $x=0$ doesn't distinguish them. Where do the fitted coefficients of $x^2$ and $x$ belong in the equation?

Step-by-step

Approach 1: Fit the quadratic in Desmos
Step 1
Enter the paired scores
Enter token counts $0,1,2$ in Desmos's $x_1$ table column and bonus scores $2,5,4$ in $y_1$ . Keep each input and its score in the same row. The table now shows the pairs $(0,2)$ , $(1,5)$ , and $(2,4)$ .
Step 2
Fit the three coefficients
The bonus is a quadratic, so use the form $ax^2+bx+c$ . Type $y_1\sim ax_1^2+bx_1+c$ ; the tilde asks Desmos to fit $a$ , $b$ , and $c$ to the table. Under PARAMETERS, Desmos shows $a=-2$ , $b=5$ , and $c=2$ . The negative $a$ also agrees with the graph opening downward.
Step 3
Match the equation to the bonus
All four choices have the same constant term, so $f(0)=2$ alone can't decide the answer. Match each fitted coefficient to its own power of $x$ . Put $-2$ with $x^2$ , $5$ with $x$ , and $2$ at the end: $f(x)=-2x^2+5x+2$ . This equation gives the game's bonus score. Choice A.

Approach 2: Use the changes between scores
Step 1
Find how the scores change
The token count rises by $1$ between rows, so compare consecutive scores. From the first two rows: $5-2=3$ From the next two rows: $4-5=-1$ The changes are not equal, so the first change isn't by itself the coefficient of $x$ .
Step 2
Connect those changes to the squared term
For a quadratic $f(x)=ax^2+bx+c$ , inputs $0,1,2$ give outputs $c$ , $a+b+c$ , and $4a+2b+c$ . Subtract adjacent outputs to find the changes: $f(1)-f(0)=a+b$ $f(2)-f(1)=3a+b$ Subtract those two changes: $(3a+b)-(a+b)=2a$ So the change between score changes equals $2a$ .
Step 3
Find the squared-term coefficient
Use the score changes $3$ and $-1$ . Subtract them: $-1-3=-4$ Set this equal to $2a$ : $2a=-4$ Divide by $2$ : $a=-2$ The squared-term coefficient is negative, as the downward-opening graph suggests.
Step 4
Find the constant term
At $x=0$ , both terms containing $x$ vanish: $f(0)=c$ The table says $f(0)=2$ , so the constant term is $c=2$
Step 5
Use one score to finish the equation
The row $x=1$ , $f(1)=5$ finds the remaining linear coefficient $b$ . Substitute $a=-2$ and $c=2$ : $5=-2(1)^2+b(1)+2$ Simplify: $5=-2+b+2$ Cancel $-2$ with $+2$ : $5=b$ So $b=5$ , and the bonus equation is $f(x)=-2x^2+5x+2$ . Choice A.

Question 7·SAT Desmos Regression Practice·Advanced Math

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