Question 15·SAT Desmos Regression Practice·Advanced Math

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A quadratic function $f$ is represented by the table.

$x$	$1$	$2$	$6$	$7$
$f(x)$	$15$	$6$	$6$	$15$

The equation $f(x)=0$ has two real solutions, $r$ and $s$ . What is the value of $rs$ ?

Your answer

A quadratic function has an $x^2$ term. For a table of its input-output pairs, Desmos quadratic regression fits the coefficients in $ax^2+bx+c$ . If the question asks for the product of the roots, the inputs that make the output $0$ , use $c/a$ instead of finding both roots. Equal outputs in the table are not necessarily zeros.

Hints

Hint 1
Put each input and its corresponding output in one row of a Desmos table. Which column should contain the $x$ -values, and which should contain the values of $f(x)$ ?
Hint 2
A quadratic regression fits an equation of the form $ax^2+bx+c$ to the table. The tilde in $y_1\sim ax_1^2+bx_1+c$ tells Desmos to find the coefficients.
Hint 3
The roots are inputs that make $f(x)=0$ . Their product comes from the constant term $c$ divided by the coefficient $a$ of $x^2$ ; you don't need to find either root.

Step-by-step

Approach 1: Fit the quadratic and use its coefficients
Step 1
Enter the paired values
Put the $x$ -values in Desmos's $x_1$ column and the corresponding $f(x)$ -values in $y_1$ . Keep each pair in one row, so the first row is $(1,15)$ and the last is $(7,15)$ . The table now shows four pairs; the $15$ s are outputs, not zeros.
Step 2
Find the quadratic's coefficients
Type $y_1\sim ax_1^2+bx_1+c$ . The tilde runs a regression, fitting the equation to the table. Under PARAMETERS, Desmos shows $a=1.8$ , $b=-14.4$ , and $c=27.6$ . Here $a$ multiplies $x^2$ , while $c$ is the constant term, with no $x$ .
Step 3
Connect the product to the coefficients
The solutions $r$ and $s$ are roots, so $f(x)=a(x-r)(x-s)$ . Set $x=0$ , where $f(0)=c$ :
$c=a(-r)(-s).$
Multiply the two negatives:
$c=ars.$
Divide by $a$ :
$rs=\frac{c}{a}.$
So divide the constant term by the coefficient of $x^2$ , rather than using the middle coefficient.
Step 4
Calculate the requested product
Type $c/a$ below the regression so Desmos uses its stored coefficients. It prints $15.333\ldots$ ; tap the fraction button to see $46/3$ . That's the product $rs$ of the two solutions. Grid in 46/3.

Approach 2: Use the table's symmetry
Step 1
Find the mirror line
The inputs $1$ and $7$ both give $15$ , and $2$ and $6$ both give $6$ . Each pair sits equally far from $4$ , so the axis of symmetry, the parabola's vertical mirror line, is $x=4$ .
Step 2
Write equations from two table entries
A quadratic mirrored across $x=4$ has vertex form $f(x)=a(x-4)^2+k$ . The distances from $4$ at inputs $1$ and $2$ are $3$ and $2$ , so their squared distances are $9$ and $4$ . Use their outputs:
$15=9a+k,$ $6=4a+k.$
Step 3
Find how the quadratic scales
Subtract the second equation from the first, so $k$ cancels:
$15-6=(9a+k)-(4a+k).$
Simplify:
$9=5a.$
Type $9\sim5x_1$ ; Desmos fits $x_1$ rather than making a slider. Under PARAMETERS it reports $x_1=1.8$ , so $a=9/5$ .
Step 4
Find the value on the mirror line
Put $a=9/5$ into the equation from $f(2)=6$ :
$6=4\left(\frac95\right)+k.$
Subtract $4(9/5)$ :
$k=6-4\left(\frac95\right).$
Type that last expression. Desmos prints $-1.2$ ; its fraction button shows $-6/5$ , so $k=-6/5$ .
Step 5
Find the roots' squared distance from 4
At a root, the output is $0$ . Set the vertex form equal to $0$ :
$0=\frac95(x-4)^2-\frac65.$
Add $6/5$ :
$\frac65=\frac95(x-4)^2.$
Divide by $9/5$ :
$(x-4)^2=\frac{\frac65}{\frac95}.$
Type the fraction on the right; Desmos prints $0.666\ldots$ . Tap its fraction button to see $2/3$ . So each root's squared distance from $4$ is $2/3$ .
Step 6
Multiply the mirror-image roots
Write the roots as $4-d$ and $4+d$ , where $d^2=2/3$ . Write their product:
$rs=(4-d)(4+d).$
Multiply using the difference of squares:
$rs=16-d^2.$
Substitute $d^2=2/3$ :
$rs=16-\frac23.$
Type $16-2/3$ ; Desmos prints $15.333\ldots$ . Tap its fraction button to see $46/3$ . The mirror line $4$ is their midpoint, not either root. Grid in 46/3.

Question 15·SAT Desmos Regression Practice·Advanced Math

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