Question

question 5 of 10
which of the following functions best describes this graph?
graph
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a. $y = (x - 2)(x - 6)$
b. $y = x^2 + 8x + 12$
c. $y = (x - 4)(x - 4)$
d. $y = x^2 - 2x + 6$

Explanation:

Step1: Analyze the graph's features

The graph is a parabola opening upwards (since the coefficient of \(x^2\) will be positive). Let's analyze each option.

Step2: Analyze Option A

\(y=(x - 2)(x - 6)=x^2-8x + 12\). The roots are at \(x = 2\) and \(x=6\) (both positive), but the graph in the image has a root (or vertex) in the negative \(x\)-region (left of \(y\)-axis), so A is incorrect.

Step3: Analyze Option B

\(y=x^2 + 8x+12=(x + 2)(x + 6)\). The roots are at \(x=-2\) and \(x = - 6\) (both negative), and the parabola opens upwards (coefficient of \(x^2\) is 1, positive). Let's check the vertex. The \(x\)-coordinate of the vertex of \(y=ax^2+bx + c\) is \(-\frac{b}{2a}\). For \(y=x^2+8x + 12\), \(a = 1\), \(b = 8\), so \(x=-\frac{8}{2\times1}=-4\). Then \(y=(-4)^2+8\times(-4)+12=16-32 + 12=-4\). So the vertex is at \((-4,-4)\), which is in the third quadrant (negative \(x\) and negative \(y\) region near the left side), matching the graph's shape (opening upwards, vertex left of \(y\)-axis).

Step4: Analyze Option C

\(y=(x - 4)(x - 4)=x^2-8x + 16\). The root is at \(x = 4\) (double root), which is positive, so the vertex is at \(x = 4\), which is in the positive \(x\)-region, not matching the graph. So C is incorrect.

Step5: Analyze Option D

\(y=x^2-2x + 6\). The discriminant \(D=b^2-4ac=(-2)^2-4\times1\times6=4 - 24=-20<0\), so no real roots (the graph doesn't cross the \(x\)-axis), but the given graph seems to have a minimum (vertex) and might cross or be near the \(x\)-axis? Wait, no, the graph in the image has a vertex below the \(x\)-axis? Wait, no, the graph in the image: looking at the axes, the \(y\)-axis has positive and negative, and the vertex is in the negative \(x\)-region. But for D, since discriminant is negative, the parabola is above the \(x\)-axis, so D is incorrect.

Answer:

B. \(y = x^2 + 8x + 12\)