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

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

Answer 1

To create a quadratic model for the dataset, we can use the given x-values and y-values to find a quadratic equation of the form \( f(x) = ax^2 + bx + c \).

We have the following points from the table:

- (-1, 6)
- (0, 5)
- (1, 6)
- (2, 9)
- (3, 14)

We can start by observing how the y-values change with respect to the x-values.

At \( x = 0 \), the y-value is 5, which suggests that when \( x = 0 \), \( f(0) = c = 5 \).

Next, we can calculate the first differences (the differences between consecutive y-values):

- From \( 5 \) to \( 6 \) (at \( x = 1 \)): \( 6 - 5 = 1 \)
- From \( 6 \) to \( 9 \) (at \( x = 2 \)): \( 9 - 6 = 3 \)
- From \( 9 \) to \( 14 \) (at \( x = 3 \)): \( 14 - 9 = 5 \)

Now, let's calculate the second differences (the differences of the first differences):

- From \( 1 \) to \( 3 \): \( 3 - 1 = 2 \)
- From \( 3 \) to \( 5 \): \( 5 - 3 = 2 \)

The second differences are constant and equal to \( 2 \), which confirms that a quadratic function is appropriate for this data set.

From the value of the second differences, we can conclude that the coefficient \( a \) in our quadratic equation \( ax^2 + bx + c \) can be determined as follows:

Since the second difference is \( 2 \), we have \( 2a = 2 \) => \( a = 1 \).

So we can start building our quadratic formula with \( a = 1 \):

Now, we already know \( c = 5 \):

Thus our equation starts as:

\[ f(x) = x^2 + bx + 5 \]

To find \( b \), we can use any of the given points. Let's use the point (1, 6):

\[
f(1) = 1^2 + b(1) + 5 = 6
\]
\[
1 + b + 5 = 6
\]
\[
b + 6 = 6
\]
\[
b = 0
\]

So we have determined the quadratic function:

\[ f(x) = x^2 + 0x + 5 = x^2 + 5 \]

So, the model we arrived at is:

**f(x) = x² + 5**

Thus, the correct response from your options provided would be:

**f(x)=x²+5**