Question

Match the equations to the graph. Watch the increments on the x & y axis

y = 1/2 (x -4)^2 -8
y = ( x -4)^2 -4
y = 2(x -4)^2 -8

Answer 1

To match the given equations to their corresponding graphs, let's analyze the equations:

1. **Equation 1: \( y = \frac{1}{2} (x - 4)^2 - 8 \)**
- This is a parabola that opens upwards with a vertex at (4, -8).
- The coefficient \( \frac{1}{2} \) indicates that it is a wider parabola.

2. **Equation 2: \( y = (x - 4)^2 - 4 \)**
- This is also a parabola that opens upwards with a vertex at (4, -4).
- The coefficient is 1, so it is a standard width parabola.

3. **Equation 3: \( y = 2(x - 4)^2 - 8 \)**
- This is another upward-opening parabola with a vertex at (4, -8).
- The coefficient 2 indicates that it is a narrower parabola compared to the other two.

### Matching Summary:
- **Vertex Locations**:
- Both Equation 1 ( \( y = \frac{1}{2} (x - 4)^2 - 8 \) ) and Equation 3 ( \( y = 2(x - 4)^2 - 8 \) ) have the same vertex at (4, -8).
- Equation 2 ( \( y = (x - 4)^2 - 4 \) ) has a vertex at (4, -4).

### Graph Characteristics:
1. Graph of \( y = 2(x - 4)^2 - 8 \): Narrow parabola (steep).
2. Graph of \( y = (x - 4)^2 - 4 \): Standard width parabola (more open).
3. Graph of \( y = \frac{1}{2} (x - 4)^2 - 8 \): Wider parabola.

You should identify the graphs based on the above characteristics of the shapes and vertices. If you have specific graphs to refer to, you can match them according to these descriptions.