To identify the equivalent equations and determine which one is most useful for identifying the extreme value, let's closely examine each equation.
First, let's rewrite each equation in its expanded form:
- Equation A: [tex]\( y = 3x^2 - 6x + 21 \)[/tex]
- Equation B: [tex]\( y = 3x^2 - 6x + 18 \)[/tex]
- Equation C: [tex]\( y = 3(x - 1)^2 + 18 \)[/tex]
- Equation D: [tex]\( y = 3(x - 1)^2 + 21 \)[/tex]
To understand the relationship between these equations, let's expand Equation C and Equation D:
- Expanding Equation C:
[tex]\[
y = 3(x - 1)^2 + 18 = 3(x^2 - 2x + 1) + 18 = 3x^2 - 6x + 3 + 18 = 3x^2 - 6x + 21
\][/tex]
- Expanding Equation D:
[tex]\[
y = 3(x - 1)^2 + 21 = 3(x^2 - 2x + 1) + 21 = 3x^2 - 6x + 3 + 21 = 3x^2 - 6x + 24
\][/tex]
Now compare the expanded forms with the given equations:
- Equation A's expanded form is [tex]\( y = 3x^2 - 6x + 21 \)[/tex].
- Equation C expands to [tex]\( y = 3x^2 - 6x + 21 \)[/tex].
Thus, Equations A and C are equivalent.
Now, to identify the equation most useful for identifying the extreme value (which is the minimum or maximum point of the quadratic function), we should look for the vertex form.
- Equation C is in the form [tex]\( y = 3(x - 1)^2 + 18 \)[/tex], which is the vertex form. The vertex form [tex]\( y = a(x-h)^2 + k \)[/tex] directly gives the vertex [tex]\( (h, k) \)[/tex], which makes it easy to identify the extreme value.
Therefore, equations A and C are equivalent, and of those, equation C is in the form most useful for identifying the extreme value of the function it defines.
So the completed statement is:
Equations [tex]\( \text{A and C} \)[/tex] are equivalent, and of those, equation [tex]\( \text{C} \)[/tex] is in the form most useful for identifying the extreme value of the function it defines.
