Question 294·Medium·Math

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A decorative arch is modeled by a quadratic function $f$ , where $f(x)$ gives the height (in feet) of the arch above the floor at a point $x$ feet from the left end.

The arch touches the floor at $x=0$ and $x=12$ , and it is $9$ feet high at $x=3$ .

Which choice gives a possible equation for $f(x)$ ?

An arch touching the floor at two inputs gives a quadratic's zeros, the inputs where its height is $0$ . Factored form builds those zeros into the equation. Use a Desmos regression on the height at a third input to find the remaining multiplier. The extra height fixes the multiplier; matching only the floor contacts can leave you with the wrong arch.

Hints

Hint 1
A zero is an input where the function's output is $0$ . At each floor contact, the arch's height is $0$ . Which two inputs are zeros of $f$ ?
Hint 2
In factored form, a zero at $r$ contributes a factor $(x-r)$ because that factor becomes $0$ at $x=r$ . Put a multiplier $a$ in front: the zeros alone don't fix the height.
Hint 3
The height at $x=3$ means input $3$ gives output $9$ . Put those values into your factored equation to find $a$ . An endpoint won't work because its output is already $0$ regardless of $a$ .

Step-by-step

Use the floor contacts and one height
Step 1
Identify the zeros
The arch touches the floor at $x=0$ and $x=12$ . A zero is an input where the height is $0$ , so:
$f(0)=0,\qquad f(12)=0.$
Step 2
Build a function with those zeros
Use factored form, $a(x-r)(x-s)$ : each factor becomes $0$ at one of the zeros. Put in $0$ and $12$ :
$f(x)=a(x-0)(x-12).$
Remove the zero inside the first factor:
$f(x)=ax(x-12).$
Keep $a$ as an unknown multiplier. The floor contacts don't tell you how tall the arch gets.
Step 3
Turn the given height into an equation
At $x=3$ , the arch is $9$ feet high, so $f(3)=9$ . Put $3$ into your factored function and set the output to $9$ :
$9=a(3)(3-12).$
Don't use a floor contact to find $a$ : putting in either zero would give $0=0$ .
Step 4
Find the multiplier in Desmos
Type $9\sim a(3)(3-12)$ in Desmos. The $\sim$ tells it to fit $a$ rather than make a slider. Under PARAMETERS, Desmos reports $a\approx-0.333333$ . Its negative sign agrees with the downward-opening curve.
Step 5
Read the exact multiplier
Type $a$ on the next line and tap the fraction button beside its printed value. Desmos shows $a=-\frac{1}{3}$ , the exact multiplier rather than a rounded decimal.
Step 6
Write the arch's equation
Replace $a$ in the factored function with $-\frac{1}{3}$ :
$f(x)=-\frac{1}{3}x(x-12).$
It has both floor contacts and gives the required height at $x=3$ . Choice A.

Question 294·Medium·Math

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