Question

the graph of the function ( f(x) = x^2 + bx + 12 ) is shown. which statements describe the graph? check all that apply.

• the vertex is the maximum value.
• the axis of symmetry is ( x = -4 ).
• the domain is all real numbers.
• the range is all real numbers.
• the function is increasing over ( (-infty, -4) ).
• the ( x )-intercepts are at ( (-6, 0) ) and ( (-2, 0) ).

Explanation:

Step1: Analyze the parabola's direction

The function is \( f(x)=x^{2}+Bx + 12 \), the coefficient of \( x^{2} \) is \( 1>0 \), so the parabola opens upward. So the vertex is the minimum value, not maximum. So "The vertex is the maximum value" is wrong.

Step2: Find the axis of symmetry

For a parabola \( y = ax^{2}+bx + c \), the axis of symmetry is \( x=-\frac{b}{2a} \). Also, from the graph, the vertex is at \( x=- 4 \) (mid - point of \( x=-6 \) and \( x = - 2 \)), so the axis of symmetry is \( x=-4 \), this statement is correct.

Step3: Determine the domain

For a quadratic function \( y=x^{2}+Bx + 12 \), there are no restrictions on the input \( x \), so the domain is all real numbers, this statement is correct.

Step4: Determine the range

Since the parabola opens upward, the range is \( y\geq k \) (where \( k \) is the \( y \) - coordinate of the vertex), not all real numbers. So "The range is all real numbers" is wrong.

Step5: Analyze the increasing/decreasing interval

Since the parabola opens upward and the axis of symmetry is \( x = - 4 \), the function is decreasing over \( (-\infty,-4) \) and increasing over \( (-4,\infty) \). So "The function is increasing over \( (-\infty,-4) \)" is wrong.

Step6: Find the x - intercepts

We can factor the quadratic function. If the roots are \( x=-6 \) and \( x=-2 \), then \( f(x)=(x + 6)(x + 2)=x^{2}+8x + 12 \), which matches the form \( f(x)=x^{2}+Bx + 12 \) (here \( B = 8 \)). Also, from the graph, the parabola intersects the \( x \) - axis at \( (-6,0) \) and \( (-2,0) \), so this statement is correct.

Answer:

• The axis of symmetry is \( x=-4 \)
• The domain is all real numbers.
• The \( x \) - intercepts are at \( (-6,0) \) and \( (-2,0) \)