Answered

Given the quadratic equation:

[tex]\[ y = a(x+3)(x-2) \][/tex]

In the equation above, [tex]\( a \)[/tex] is a nonzero constant. The graph of the equation in the [tex]\( xy \)[/tex]-plane is a parabola with vertex [tex]\( (h, k) \)[/tex].

Which of the following is equal to [tex]\( h \)[/tex], in terms of [tex]\( a \)[/tex]?

A. [tex]\( -\frac{1}{2} \)[/tex]

B. [tex]\( 0 \)[/tex]

C. [tex]\( \frac{1}{2} \)[/tex]

D. [tex]\( 2 \)[/tex]



Answer :

GinnyAnswer
To find the x-coordinate of the vertex [tex]\( h \)[/tex] of a quadratic equation given in the form [tex]\( y = a(x + b)(x + c) \)[/tex], we can follow these steps:

1. Identify the Roots of the Quadratic Equation:
The given equation is [tex]\( y = a(x + 3)(x - 2) \)[/tex]. The roots (or zeros) of the equation occur where [tex]\( y = 0 \)[/tex]. This happens when each factor is zero.
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]
[tex]\[ x - 2 = 0 \implies x = 2 \][/tex]

2. Calculate the Midpoint of the Roots:
The x-coordinate of the vertex [tex]\( h \)[/tex] is the midpoint of the roots [tex]\( -3 \)[/tex] and [tex]\( 2 \)[/tex]. The midpoint formula for two points [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] is:
[tex]\[ h = \frac{x_1 + x_2}{2} \][/tex]
Substituting the roots [tex]\( x_1 = -3 \)[/tex] and [tex]\( x_2 = 2 \)[/tex]:
[tex]\[ h = \frac{-3 + 2}{2} \][/tex]

3. Simplify the Expression:
Simplifying the fraction:
[tex]\[ h = \frac{-3 + 2}{2} = \frac{-1}{2} = -0.5 \][/tex]

Thus, the x-coordinate of the vertex [tex]\( h \)[/tex] is [tex]\( -0.5 \)[/tex]. Therefore, the answer is:

[tex]\[ (A) \ - \frac{1}{2} \][/tex]