Question

Match the equation with its feature.(4 points)
y=−2(x+3)2−7

y=1/2(x+3)2−7

y=(x+7)2−3

y=(x−3)2+7

Answer 1

To match the equations with their features, we'll analyze each equation based on standard features of quadratic functions, specifically their vertex form \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

1. **\(y = -2(x + 3)^2 - 7\)**:
- This equation has \(a = -2\) (the parabola opens downwards) and the vertex at \((-3, -7)\).

2. **\(y = \frac{1}{2}(x + 3)^2 - 7\)**:
- Here, \(a = \frac{1}{2}\) (the parabola opens upwards and is wider), and the vertex at \((-3, -7)\).

3. **\(y = (x + 7)^2 - 3\)**:
- In this case, \(a = 1\) (the parabola opens upwards) and has the vertex at \((-7, -3)\).

4. **\(y = (x - 3)^2 + 7\)**:
- This has \(a = 1\) (the parabola opens upwards) and the vertex at \((3, 7)\).

Now, let's summarize the features:

1. **Parabola Open Downwards, Vertex (-3, -7)**:
- \(y = -2(x + 3)^2 - 7\)

2. **Parabola Open Upwards, Vertex (-3, -7)**:
- \(y = \frac{1}{2}(x + 3)^2 - 7\)

3. **Parabola Open Upwards, Vertex (-7, -3)**:
- \(y = (x + 7)^2 - 3\)

4. **Parabola Open Upwards, Vertex (3, 7)**:
- \(y = (x - 3)^2 + 7\)

So, if you need the features matched explicitly to the equations based on their vertex and whether they open upwards or downwards, they would be:

- **\(y = -2(x + 3)^2 - 7\)**: Parabola opens downwards, vertex (-3, -7)
- **\(y = \frac{1}{2}(x + 3)^2 - 7\)**: Parabola opens upwards, vertex (-3, -7)
- **\(y = (x + 7)^2 - 3\)**: Parabola opens upwards, vertex (-7, -3)
- **\(y = (x - 3)^2 + 7\)**: Parabola opens upwards, vertex (3, 7)