To solve the problem, we need to write the quadratic equations in vertex form for each graph provided. 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:
- If \( a > 0 \), the parabola opens upwards.
- If \( a < 0 \), the parabola opens downwards.
- The absolute value of \( a \) affects the width: larger \( |a| \) makes the parabola narrower, and smaller \( |a| \) makes it wider.
Let's analyze each graph step by step:
---
Graph 1
-
Vertex: The vertex is at \( (0, 0) \).
-
Direction: The parabola opens upwards.
-
Width: The parabola appears to have a standard width, so \( a = 1 \).
The equation is:
\[
y = 1(x - 0)^2 + 0 \quad \text{or simply} \quad y = x^2
\]
---
Graph 2
-
Vertex: The vertex is at \( (0, 0) \).
-
Direction: The parabola opens upwards.
-
Width: The parabola appears to be narrower than the standard parabola, so \( a > 1 \). From the graph, it looks like \( a = 2 \).
The equation is:
\[
y = 2(x - 0)^2 + 0 \quad \text{or simply} \quad y = 2x^2
\]
---
Graph 3
-
Vertex: The vertex is at \( (2, -3) \).
-
Direction: The parabola opens upwards.
-
Width: The parabola appears to have a standard width, so \( a = 1 \).
The equation is:
\[
y = 1(x - 2)^2 - 3 \quad \text{or simply} \quad y = (x - 2)^2 - 3
\]
---
Graph 4
-
Vertex: The vertex is at \( (-1, 2) \).
-
Direction: The parabola opens upwards.
-
Width: The parabola appears to have a standard width, so \( a = 1 \).
The equation is:
\[
y = 1(x + 1)^2 + 2 \quad \text{or simply} \quad y = (x + 1)^2 + 2
\]
---
Graph 5
-
Vertex: The vertex is at \( (0, 4) \).
-
Direction: The parabola opens downwards.
-
Width: The parabola appears to have a standard width, so \( a = -1 \).
The equation is:
\[
y = -1(x - 0)^2 + 4 \quad \text{or simply} \quad y = -x^2 + 4
\]
---
Graph 6
-
Vertex: The vertex is at \( (-2, -1) \).
-
Direction: The parabola opens downwards.
-
Width: The parabola appears to have a standard width, so \( a = -1 \).
The equation is:
\[
y = -1(x + 2)^2 - 1 \quad \text{or simply} \quad y = -(x + 2)^2 - 1
\]
---
Final Answers
1. \( y = x^2 \)
2. \( y = 2x^2 \)
3. \( y = (x - 2)^2 - 3 \)
4. \( y = (x + 1)^2 + 2 \)
5. \( y = -x^2 + 4 \)
6. \( y = -(x + 2)^2 - 1 \)
\[
\boxed{
\begin{aligned}
1. & \ y = x^2 \\
2. & \ y = 2x^2 \\
3. & \ y = (x - 2)^2 - 3 \\
4. & \ y = (x + 1)^2 + 2 \\
5. & \ y = -x^2 + 4 \\
6. & \ y = -(x + 2)^2 - 1
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of graphing quadratic functions in vertex form worksheet.
