Answer:
(a) f(x) = -x² -6x +7
Explanation:
You want the standard-form equation of the function graphed.
Observation
The first thing you notice is that the parabola opens downward. Also relevant is the location of the vertex at (-3, 16) and/or the x-intercepts at -7 and +1. These give you several ways to create or identify the equation.
Paring the choices
The fact that the graph opens downward means the leading coefficient (of x²) is negative. That eliminates choices C and D. The remaining choices are different only in their sign of the 6x term. Their leading coefficient is -1.
Equation from vertex
The vertex form of the equation of this parabola is ...
f(x) = a(x -h)² +k . . . . . . . . for leading coefficient 'a' and vertex (h, k)
For leading coefficient -1, and vertex (h, k) = (-3, 16), the equation is ...
f(x) = -(x -(-3))² +16
f(x) = -(x² +6x +9) +16
f(x) = -x² -6x +7 . . . . . . . . matches choice A
Equation from intercepts
For x-intercepts p and q, the factored form of the equation of the parabola is ...
f(x) = a(x -p)(x -q) . . . . . a=leading coefficient, as above
f(x) = -(x -(-7))(x -1) = -(x +7)(x -1) = -(x² +6x -7)
f(x) = -x² -6x +7 . . . . . . . . matches choice A
Choice from line of symmetry
The line of symmetry of the graph of y = ax² +bx +c is ...
x = -b/(2a)
For the two remaining answer choices, the line of symmetry would be ...
(a) x = -(-6)/(2(-1)) = -3 . . . . . . odd number of minus signs, so negative
(b) x = -(6)/(2(-1)) = 3 . . . . . . . even number of minus signs, so positive
Clearly, the line of symmetry is x = -3, so answer choice A is appropriate.
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