Question

here is the graph of a quadratic function y. determine whether each equation below could define the function f
a. $f(x) = -x^2 + 8x - 12$
yes
no
b. $f(x) = x^2 - 8x + 12$
yes
no
c. $f(x) = (x + 2)(x + 6)$
yes
no
d. $f(x) = (x - 2)(x - 6)$
yes
no
the graph has x - intercepts at (2, 0) and (6, 0), and vertex at (4, - 4).

Explanation:

Step1: Analyze the parabola's direction and roots

The parabola opens upwards (since it has a minimum point), so the coefficient of \(x^2\) should be positive. The roots are at \(x = 2\) and \(x = 6\), so the factored form should be \(f(x)=(x - 2)(x - 6)\) or expanded form with positive leading coefficient.

Step2: Check Option A

For \(f(x)=-x^2 + 8x - 12\), the leading coefficient is \(-1\) (negative), so the parabola opens downward. But our graph opens upward. So A is No.

Step3: Check Option B

Expand \(f(x)=x^2 - 8x + 12\). Factor it: \(x^2 - 8x + 12=(x - 2)(x - 6)\) (since \(2\times6 = 12\) and \(2 + 6 = 8\)). The leading coefficient is \(1\) (positive), opens upward. Let's check the vertex. The vertex of \(ax^2+bx+c\) is at \(x = -\frac{b}{2a}\). Here, \(a = 1\), \(b=-8\), so \(x = \frac{8}{2}=4\). Then \(f(4)=16 - 32 + 12=-4\), which matches the vertex \((4, -4)\). So B is Yes.

Step4: Check Option C

\(f(x)=(x + 2)(x + 6)=x^2 + 8x + 12\). The roots would be at \(x=-2\) and \(x=-6\), but our roots are at \(x = 2\) and \(x = 6\). So C is No.

Step5: Check Option D

\(f(x)=(x - 2)(x - 6)\) (assuming the typo is corrected, as it should be \((x - 2)(x - 6)\)). This is the factored form with roots at \(x = 2\) and \(x = 6\), leading coefficient \(1\) (positive), opens upward. Expanding it gives \(x^2 - 8x + 12\), which matches Option B. So D is Yes (assuming the typo in the original is a misprint, and it's \((x - 2)(x - 6)\)). Wait, but let's re - check: If D is \((x - 2)(x - 6)\), then it's the same as the factored form. So:

• A: No (downward opening)
• B: Yes (expanded form, vertex matches)
• C: No (wrong roots)
• D: Yes (factored form, roots match, positive leading coefficient)

Wait, let's re - expand B: \(x^2-8x + 12\), when \(x = 4\), \(f(4)=16-32 + 12=-4\), which is the vertex. For D, \((x - 2)(x - 6)=x^2-8x + 12\), same as B. So:

• A: No
• B: Yes
• C: No
• D: Yes (assuming the typo in D's original is a mistake, and it's \((x - 2)(x - 6)\))

Answer:

• A: NO
• B: YES
• C: NO
• D: YES (assuming the typo in D is corrected to \((x - 2)(x - 6)\))

(Note: If D was supposed to be \((x - 2)(x - 6)\) as per the roots, then D is Yes. If there was a different typo, but based on the graph, the correct equations are B and D (since B is the expanded form of D's factored form) and A and C are incorrect.)