Question

Create an equation that models the transformation of y = 2x*2 to the function in the graph
Upward u shape with points (0,-4), (-1,-2), (1,-2),(-2,4),(2,4)
y = 2(x-4)^2
y = 2(x+4)^2
y = 2x^2-4
y= 2x^2 +4

Answer 1

To find the correct equation that matches the given points \((0, -4)\), \((-1, -2)\), \((1, -2)\), \((-2, 4)\), and \((2, 4)\), we need to start with the transformation of the function \(y = 2x^2\).

1. **Identify the vertex form and its transformations**: The standard upward opening parabola is \(y = 2(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

2. **Determine the vertex**: From the points given, particularly \((0, -4)\), it seems the vertex might be at \((0, -4)\).

3. **Vertical Shift**: Since the graph appears to open upwards and the lowest point (vertex) is \(-4\), we can infer that the entire parabola is shifted downwards by \(4\) units.

4. **Building the equation**: The equation then becomes:

\[
y = 2x^2 - 4
\]

5. **Verification**: Let's check if this equation passes through the given points.
- For \((0, -4)\):
\[
y = 2(0)^2 - 4 = -4 \quad \text{(correct)}
\]
- For \((-1, -2)\):
\[
y = 2(-1)^2 - 4 = 2(1) - 4 = -2 \quad \text{(correct)}
\]
- For \((1, -2)\):
\[
y = 2(1)^2 - 4 = 2(1) - 4 = -2 \quad \text{(correct)}
\]
- For \((-2, 4)\):
\[
y = 2(-2)^2 - 4 = 2(4) - 4 = 8 - 4 = 4 \quad \text{(correct)}
\]
- For \((2, 4)\):
\[
y = 2(2)^2 - 4 = 2(4) - 4 = 8 - 4 = 4 \quad \text{(correct)}
\]

Since all calculations confirm that the points fit this equation, the correct transformation for the function to match the graph with the specified points is:

\[
\boxed{y = 2x^2 - 4}
\]