IM2 Fall Sem Exam

Question

4 of 204 of 20 Items

Question
Which table represents a quadratic function?
Responses

x k(x)
-2 2
-1 1
0 0
1 1
2 2


x k(x) -2 2 -1 1 0 0 1 1 2 2

x h(x)
-2

-1 1
0 3
1 9
2 27


x h(x) -2 1 third -1 1 0 3 1 9 2 27

x g(x)
-2 0
-1 1
0 4
1 9
2 16


x g(x) -2 0 -1 1 0 4 1 9 2 16

x f(x)
-2 -3
-1 0
0 3
1 6
2 9

Answer 1

To determine which table represents a quadratic function, we need to check the relationship between the values of \(x\) and \(k(x)\), \(h(x)\), \(g(x)\), and \(f(x)\) in each table. A quadratic function can be identified because the second differences of the output values (\(k(x)\), \(h(x)\), \(g(x)\), \(f(x)\)) will be constant.

Let's analyze the given tables one at a time.

1. Table for \(k(x)\):

   • Values: \(2, 1, 0, 1, 2\) (for \(x = -2, -1, 0, 1, 2\))
   • First differences: \(-1, -1, 1, 1\)
   • Second differences: \(0, 0, 0\) (not constant)

2. Table for \(h(x)\):

   • Values: \(1, 3, 9, 27\)
   • First differences: \(2, 6, 18\)
   • Second differences: \(4, 12\) (not constant)

3. Table for \(g(x)\):

   • Values: \(0, 1, 4, 9, 16\) (for \(x = -2, -1, 0, 1, 2\))
   • First differences: \(1, 3, 5, 7\)
   • Second differences: \(2, 2, 2\) (constant)

4. Table for \(f(x)\):

   • Values: \(-3, 0, 3, 6, 9\)
   • First differences: \(3, 3, 3, 3\)
   • Second differences: \(0, 0, 0\) (not constant)

From this analysis, only the table for \(g(x)\) has constant second differences, implying it represents a quadratic function.

Therefore, the table that represents a quadratic function is the one for \(g(x)\):

\[ \begin{array}{c|c} x & g(x) \ \hline -2 & 0 \ -1 & 1 \ 0 & 4 \ 1 & 9 \ 2 & 16 \ \end{array} \]