To rewrite a quadratic expression of the form [tex]\( y = ax^2 + bx + c \)[/tex] in the vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], certain properties must hold true. Let's break down the process of converting the standard form to the vertex form and analyze each option given.
1. Identify the Vertex (h, k):
The vertex form of a quadratic function [tex]\( y = a(x - h)^2 + k \)[/tex] highlights the vertex of the parabola at [tex]\((h, k)\)[/tex].
- Finding [tex]\( h \)[/tex]:
[tex]\( h \)[/tex] is the x-coordinate of the vertex and is found by using the formula [tex]\( h = -\frac{b}{2a} \)[/tex].
- Finding [tex]\( k \)[/tex]:
Once [tex]\( h \)[/tex] is determined, [tex]\( k \)[/tex] can be found by substituting [tex]\( h \)[/tex] back into the original quadratic equation:
[tex]\[ k = c - \frac{b^2}{4a} \][/tex]
2. Convert to Vertex Form:
With [tex]\( h \)[/tex] and [tex]\( k \)[/tex] determined, rewrite the quadratic:
[tex]\[ y = a \left( x - \left( -\frac{b}{2a} \right) \right)^2 + \left( c - \frac{b^2}{4a} \right) \][/tex]
Which simplifies to:
[tex]\[ y = a (x - h)^2 + k \][/tex]
Now let's analyze the given options:
1. "h and k cannot both equal zero"
- This statement is not necessarily true. If [tex]\( b = 0 \)[/tex] and [tex]\( c = 0 \)[/tex], [tex]\( h \)[/tex] and [tex]\( k \)[/tex] both can be zero. Thus, this is not a mandatory condition.
2. "k and c have the same value"
- This is false. [tex]\( k \)[/tex] and [tex]\( c \)[/tex] do not necessarily have the same value. [tex]\( k \)[/tex] is derived from [tex]\( c \)[/tex] by the formula [tex]\( k = c - \frac{b^2}{4a} \)[/tex].
3. "The value of [tex]\( a \)[/tex] remains the same"
- This is true. The coefficient [tex]\( a \)[/tex] in the standard quadratic and its vertex form remains unchanged. The value of [tex]\( a \)[/tex] does not alter during the conversion.
4. "h is equal to one half [tex]\(-b\)[/tex]"
- This is somewhat misleading in its wording. The actual relationship is [tex]\( h = -\frac{b}{2a} \)[/tex], not simply half of [tex]\( -b \)[/tex]. Therefore, [tex]\( h \)[/tex] is not directly half of [tex]\( -b \)[/tex] unless [tex]\( a = 1 \)[/tex], which is a special case and not generally true.
Taking this analysis into account, the correct statement from the provided options is:
- The value of [tex]\( a \)[/tex] remains the same.
