Question

Use the table to answer the question.

This table shows the corresponding y-values for a given set of x-values.
x-values y-values
−1
6
0 5
1 6
2 9
3 14
Create a quadratic model for the dataset in the xy
-chart.

(1 point)
Responses

f(x)=x2+7
f left parenthesis x right parenthesis equals x squared plus 7

f(x)=x2+5
f left parenthesis x right parenthesis equals x squared plus 5

f(x)=2x+5
f left parenthesis x right parenthesis equals 2 x plus 5

f(x)=x+5
f left parenthesis x right parenthesis equals x plus 5
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Answer 1

To create a quadratic model for the dataset in the xy-chart, we need to find a quadratic function of the form \( f(x) = ax^2 + bx + c \) that fits the given data points.

The provided points are:
- \( (-1, 6) \)
- \( (0, 5) \)
- \( (1, 6) \)
- \( (2, 9) \)
- \( (3, 14) \)

### Step 1: Identify Parameters
We can substitute the known points into the quadratic equation to find \( a, b, \) and \( c \).

1. **Using point (0, 5)**:
\[
f(0) = a(0)^2 + b(0) + c = c \implies c = 5
\]

2. **Now use point (1, 6)**:
\[
f(1) = a(1)^2 + b(1) + c = 6 \implies a + b + 5 = 6 \implies a + b = 1 \quad (1)
\]

3. **Using point (2, 9)**:
\[
f(2) = a(2)^2 + b(2) + c = 9 \implies 4a + 2b + 5 = 9 \implies 4a + 2b = 4 \implies 2a + b = 2 \quad (2)
\]

### Step 2: Solve the System of Equations
Now we have a system of two equations from steps 2 and 3:
1. \( a + b = 1 \) (Equation 1)
2. \( 2a + b = 2 \) (Equation 2)

We can subtract Equation 1 from Equation 2:
\[
(2a + b) - (a + b) = 2 - 1 \implies a = 1
\]

Now we can substitute \( a = 1 \) back into Equation 1:
\[
1 + b = 1 \implies b = 0
\]

### Step 3: Construct the Function
Now we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = 5 \)

Thus, the quadratic model is:
\[
f(x) = 1x^2 + 0x + 5 = x^2 + 5
\]

### Conclusion
The quadratic model for the dataset is:
**f(x) = x² + 5**

This corresponds to the response:
**f(x) = x² + 5**