Question 24·SAT Desmos Regression Practice·Advanced Math

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A quadratic function $f$ has a positive leading coefficient. The equation $f(x)=9$ has exactly two solutions: $x=-2$ and $x=10$ . Also, $f(1)=0$ . What is the coefficient of $x$ in the standard form of $f$ ?

Your answer

Two inputs with the same quadratic output point to a shift before factoring. The solutions of $f(x)=k$ are zeros of $f(x)-k$ , meaning inputs that make it $0$ . Factor the shifted function, not the original function. Then use the extra function value in a Desmos regression to find the scale, and take only the requested coefficient. Don't mistake the given solutions for zeros of $f$ .

Hints

Hint 1
The two listed inputs both give output $9$ . Subtracting $9$ makes them zeros, or inputs that give $0$ , of the shifted function $f(x)-9$ . Which factors become $0$ at those inputs?
Hint 2
Once you have the shifted factored form, use $f(1)=0$ to find its scale. Put $1$ in for each $x$ and set the entire function equal to $0$ , not $9$ .
Hint 3
To find the coefficient of $x$ , you don't need the entire expanded function. In $(x+2)(x-10)$ , find the two products containing one $x$ . What do they add to?

Step-by-step

Factor the shifted function
Step 1
Build a function with the given solutions
Because $f(-2)=f(10)=9$ , subtracting $9$ makes both outputs $0$ . A zero is an input that gives output $0$ , so $f(x)-9$ has factors $x+2$ and $x-10$ . Write the function with the $9$ outside the factors: $f(x)=9+a(x+2)(x-10)$ . Here $a$ is the leading coefficient, the number multiplying $x^2$ after expansion.
Step 2
Turn the extra value into an equation
$f(1)=0$ says that input $1$ gives output $0$ . Substitute $1$ for each $x$ in the function you built: $0=9+a(1+2)(1-10)$ . Use $0$ as the output here; $9$ is the output at the other two inputs.
Step 3
Find the scale in Desmos
Type $0\sim9+a(1+2)(1-10)$ . The $\sim$ asks Desmos to find the unknown $a$ . Under PARAMETERS, Desmos shows $a\approx0.333333$ , which is positive as required. Keep its stored value rather than retyping the rounded decimal.
Step 4
Identify the products that make an x term
In standard form $ax^2+bx+c$ , $b$ is the coefficient of $x$ . From $(x+2)(x-10)$ , the products containing exactly one $x$ come from $x$ times $-10$ and $2$ times $x$ . Multiply those products: $x(-10)+2(x)=-10x+2x$ .
Step 5
Combine the x terms
Combine the two terms: $-10x+2x=-8x$ . The factor $a$ makes the coefficient of $x$ equal to $-8a$ . The $9$ outside the factors is a constant, so it adds no $x$ term.
Step 6
Evaluate the requested coefficient
Type $-8a$ on a new line. Desmos uses its stored value of $a$ and shows about $-2.666667$ ; the fraction button shows $-\frac{8}{3}$ . That's the coefficient of $x$ , not the leading coefficient. Grid in -8/3.

Question 24·SAT Desmos Regression Practice·Advanced Math

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