Question 234·Hard·Math

0 / 2000

The function $f$ is defined by $f(x)=(x+3)(x-c)$ , where $c$ is a positive constant. The graph of $y=f(x)$ has a minimum point that lies on the line $y=2x-25$ .

Which choice is the value of $c$ ?

A quadratic in factored form shows its zeros, the inputs where its output is $0$ . If its minimum lies on a line, find the midpoint of the zeros and work out the parabola’s output there. That output must equal the line’s output at the same input. Use a Desmos regression to solve the resulting equation, then check the positive-constant restriction.

Hints

Hint 1
A zero is an input that makes a function equal $0$ . Each factor gives one zero when you set it equal to $0$ . Watch the sign in $x+3$ .
Hint 2
The vertex is the parabola’s turning point. A parabola mirrors itself around that point, so the vertex’s input is halfway between the two zeros.
Hint 3
The minimum point sits on both graphs. Find the parabola’s output at the vertex, then set it equal to what the line gives at that same input.

Step-by-step

Find the vertex, then fit the constant
Step 1
Find the two zeros
A zero makes $f(x)=0$ , so set each factor equal to $0$ . From $x+3=0$ , you get $x=-3$ ; from $x-c=0$ , you get $x=c$ . The plus sign in $x+3$ gives a negative zero.
Step 2
Find the minimum’s input
The vertex is the parabola’s turning point. The factors start with $x$ , so the parabola opens upward and its vertex is its minimum. It mirrors itself around the vertex, so average the zeros to find its horizontal coordinate: $h=\frac{c-3}{2}$ .
Step 3
Find the minimum’s output
Put $h$ into both factors: $f(h)=\left(\frac{c-3}{2}+3\right)\left(\frac{c-3}{2}-c\right)$ . Simplify each factor: $f(h)=\frac{c+3}{2}\cdot\frac{-c-3}{2}$ . Multiply: $f(h)=-\frac{(c+3)^2}{4}$ .
Step 4
Make the vertex satisfy the line
The parabola and the line must give the same output at the vertex’s input. Translate “lies on the line”: $f(h)=2h-25$ . Replace $f(h)$ and $h$ : $-\frac{(c+3)^2}{4}=2\left(\frac{c-3}{2}\right)-25$ . Simplify the right side: $-\frac{(c+3)^2}{4}=c-28$ .
Step 5
Solve for the positive constant
Type $-\frac{(c_1+3)^2}{4}\sim c_1-28\{c_1>0\}$ . In this regression, the subscripted $c_1$ is the value Desmos fits, $\sim$ asks it to make the sides equal, and the braces enforce positive $c$ . Under PARAMETERS, Desmos shows $c_1\approx6.314$ .
Step 6
Match the exact value
Type $-5+8\sqrt{2}$ . Desmos prints about $6.314$ , matching the fitted value. The regression gave a decimal, but the choice gives the exact value of the positive constant: $c=-5+8\sqrt{2}$ . Choice A.

Question 234·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this