Use the table to answer the question.

Question

x-values y-values
−1
−
1
6
0 5
1 6
2 9
3 14
Create a quadratic model for the dataset in the xy
𝑥
𝑦
-chart.

(1 point)
Responses

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

f(x)=x2+7
𝑓
(
𝑥
)
=
𝑥
2
+
7
f left parenthesis x right parenthesis equals x squared plus 7

f(x)=x2+5
𝑓
(
𝑥
)
=
𝑥
2
+
5
f left parenthesis x right parenthesis equals x squared plus 5

f(x)=2x+5

Answer 1

To find a quadratic model for the given dataset, we can use the points provided in the table. The values appear to suggest a quadratic relationship, so we can fit the data to a quadratic function of the form \( f(x) = ax^2 + bx + c \).

The x-values and corresponding y-values are:

• For \( x = -1 \), \( y = 6 \)
• For \( x = 0 \), \( y = 5 \)
• For \( x = 1 \), \( y = 6 \)
• For \( x = 2 \), \( y = 9 \)
• For \( x = 3 \), \( y = 14 \)

We can create a system of equations using these points. However, since we are looking for a straightforward fit based on the response options, we can evaluate the given options.

1. Option 1: \( f(x) = x + 5 \)

   • For \( x = 0 \), \( f(0) = 5 \) (matches)
   • For \( x = 1 \), \( f(1) = 6 \) (matches)
   • For \( x = 2 \), \( f(2) = 7 \) (does not match)

   This option does not fit all points.

2. Option 2: \( f(x) = x^2 + 7 \)

   • For \( x = -1 \), \( f(-1) = 8 \) (does not match)

   This option does not fit all points.

3. Option 3: \( f(x) = x^2 + 5 \)

   • For \( x = -1 \), \( f(-1) = 6 \) (matches)
   • For \( x = 0 \), \( f(0) = 5 \) (matches)
   • For \( x = 1 \), \( f(1) = 6 \) (matches)
   • For \( x = 2 \), \( f(2) = 9 \) (matches)
   • For \( x = 3 \), \( f(3) = 14 \) (matches)

   This option fits all points.

4. Option 4: \( f(x) = 2x + 5 \)

   • For \( x = 0 \), \( f(0) = 5 \) (matches)
   • For \( x = 1 \), \( f(1) = 7 \) (does not match)

   This option does not fit all points.

Based on the evaluations, the quadratic model that fits the dataset is:

\( f(x) = x^2 + 5 \)