The x-intercepts of a quadratic function are 2 and -4 . Which could be the function ?
I.`y=x^2 + 2x-8`
II.`y=-2x^2-4x+16`
III.`y=(x+1)^2 -9`

Options

A. II and III only
B. I and III only
C. I and II only
D. I , II and III

Answer

To determine which quadratic function has x-intercepts at 2 and -4, we can start by using the fact that the x-intercepts (or roots) of a quadratic function can be used to form the function itself.

### Step-by-step Solution:

1. **Identify the Roots**: The x-intercepts given are 2 and -4. These correspond to the roots of the quadratic function.

2. **Form the Quadratic Function**: The general form of a quadratic function based on its roots can be expressed as:
\[ y = (x - r_1)(x - r_2) \]
where \( r_1 \) and \( r_2 \) are the roots. Here, \( r_1 = 2 \) and \( r_2 = -4 \).

3. **Substitute the Roots**: Plugging in the roots into the formula gives:
\[ y = (x - 2)(x + 4) \]

4. **Expand the Expression**:
\[ y = x^2 + 4x - 2x - 8 \]
Simplifying this, we get:
\[ y = x^2 + 2x - 8 \]

5. **Compare with Given Options**: Now we will compare this function with the given options:
- I. \( y = x^2 + 2x - 8 \) (This matches our derived function)
- II. \( y = -2x^2 - 4x + 16 \) (This does not match)
- III. \( y = (x + 1)^2 - 9 \)

6. **Simplify Option III**: Let's simplify option III:
\[ y = (x + 1)^2 - 9 \]
Expanding this gives:
\[ y = x^2 + 2x + 1 - 9 = x^2 + 2x - 8 \]
This also matches our derived function.

7. **Conclusion**: Therefore, the functions that could be the quadratic function with x-intercepts at 2 and -4 are:
- I. \( y = x^2 + 2x - 8 \)
- III. \( y = (x + 1)^2 - 9 \)

### Final Answer:
The correct functions are I and III.

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