The table shows the values of the quadratic functions f for several values of x. Which one of the following best represents f?
`{:(x,-1,0,1," "2),(f(x)," "1,3,1,-5):}`

Options

A. `f(x) = -2x^2`
B. `f(x) = x^2 + 3`
C. `f(x) = -x^2 + 3`
D. `f(x) = -2x^2 + 3`

Answer

To determine which quadratic function best represents the values given in the table, we will evaluate each option by substituting the values of \( x \) into the corresponding function and checking if the output matches the values of \( f(x) \).

### Given Table:
- \( x: -1, 0, 1, 2 \)
- \( f(x): 1, 3, 1, -5 \)

### Options to Check:
1. \( f(x) = -2x^2 \)
2. \( f(x) = 3x + 1 \)
3. \( f(x) = -x^2 + 3 \)
4. \( f(x) = -2x^2 + 3x \)

### Step-by-Step Evaluation:

#### Step 1: Check Option 1: \( f(x) = -2x^2 \)
- For \( x = -1 \):
\[ f(-1) = -2(-1)^2 = -2 \quad \text{(not equal to 1)} \]
- Since the output does not match, this option is incorrect.

**Hint:** Substitute \( x \) into the function and compare with the corresponding \( f(x) \).

#### Step 2: Check Option 2: \( f(x) = 3x + 1 \)
- For \( x = -1 \):
\[ f(-1) = 3(-1) + 1 = -3 + 1 = -2 \quad \text{(not equal to 1)} \]
- Since the output does not match, this option is incorrect.

**Hint:** Always check all given values of \( x \) for each option.

#### Step 3: Check Option 3: \( f(x) = -x^2 + 3 \)
- For \( x = -1 \):
\[ f(-1) = -(-1)^2 + 3 = -1 + 3 = 2 \quad \text{(not equal to 1)} \]
- Since the output does not match, this option is incorrect.

**Hint:** Ensure to evaluate the function for all values of \( x \) given in the table.

#### Step 4: Check Option 4: \( f(x) = -2x^2 + 3x \)
- For \( x = -1 \):
\[ f(-1) = -2(-1)^2 + 3(-1) = -2 - 3 = -5 \quad \text{(not equal to 1)} \]
- For \( x = 0 \):
\[ f(0) = -2(0)^2 + 3(0) = 0 \quad \text{(not equal to 3)} \]
- For \( x = 1 \):
\[ f(1) = -2(1)^2 + 3(1) = -2 + 3 = 1 \quad \text{(equal to 1)} \]
- For \( x = 2 \):
\[ f(2) = -2(2)^2 + 3(2) = -8 + 6 = -2 \quad \text{(not equal to -5)} \]

After checking all options, it seems none of the options perfectly match all the values in the table. However, the closest option that satisfies at least one point is option 4.

### Conclusion:
None of the given options perfectly represent the function \( f \) based on the values provided in the table.