To solve the problem, we need to identify which of the given quadratic equations are in
vertex form. The vertex form of a quadratic equation is:
\[
y = a(x - h)^2 + k
\]
where:
- \( (h, k) \) is the vertex of the parabola,
- \( a \) determines the direction and the width of the parabola.
Step-by-Step Analysis
Let's analyze each equation to determine if it is in vertex form.
#### Equation A: \( y = 3(x - 6)^2 + 2 \)
- This equation is already in the form \( y = a(x - h)^2 + k \).
- Here, \( a = 3 \), \( h = 6 \), and \( k = 2 \).
-
Conclusion: This is in vertex form.
#### Equation B: \( y = -4x^2 + 13x - 12 \)
- This equation is in standard form \( y = ax^2 + bx + c \).
- It is not in the form \( y = a(x - h)^2 + k \).
-
Conclusion: This is not in vertex form.
#### Equation C: \( y = -6(x + 2)^2 + 5 \)
- This equation is already in the form \( y = a(x - h)^2 + k \).
- Here, \( a = -6 \), \( h = -2 \), and \( k = 5 \).
-
Conclusion: This is in vertex form.
#### Equation D: \( y = 7x^2 + 20x - 8 \)
- This equation is in standard form \( y = ax^2 + bx + c \).
- It is not in the form \( y = a(x - h)^2 + k \).
-
Conclusion: This is not in vertex form.
#### Equation E: \( y = 3(x + 5)^2 - 9 \)
- This equation is already in the form \( y = a(x - h)^2 + k \).
- Here, \( a = 3 \), \( h = -5 \), and \( k = -9 \).
-
Conclusion: This is in vertex form.
#### Equation F: \( y = -8(x - 7)^2 + 8 \)
- This equation is already in the form \( y = a(x - h)^2 + k \).
- Here, \( a = -8 \), \( h = 7 \), and \( k = 8 \).
-
Conclusion: This is in vertex form.
#### Equation G: \( y = 2(x - 3)^2 + 5 \)
- This equation is already in the form \( y = a(x - h)^2 + k \).
- Here, \( a = 2 \), \( h = 3 \), and \( k = 5 \).
-
Conclusion: This is in vertex form.
#### Equation H: \( y = 3x^2 + 9x + 9 \)
- This equation is in standard form \( y = ax^2 + bx + c \).
- It is not in the form \( y = a(x - h)^2 + k \).
-
Conclusion: This is not in vertex form.
Final Answer
The equations in vertex form are:
- Equation A: \( y = 3(x - 6)^2 + 2 \)
- Equation C: \( y = -6(x + 2)^2 + 5 \)
- Equation E: \( y = 3(x + 5)^2 - 9 \)
- Equation F: \( y = -8(x - 7)^2 + 8 \)
- Equation G: \( y = 2(x - 3)^2 + 5 \)
Thus, the final answer is:
\[
\boxed{A, C, E, F, G}
\]
Parent Tip: Review the logic above to help your child master the concept of converting standard form to vertex form worksheet.
