Answer the questions below to determine what kind of function is depicted in the table below.

Question

xx minus, 1−1 00 11 22 33
f, of, xf(x) minus, 7−7 minus, 48−48 minus, 171−171 minus, 376−376 minus, 663−663
Answer
Attempt 1 out of 2

This function is
because
.

Answer 1

To analyze the function depicted in the table, we first look at the given pairs of \(x\) and \(f(x)\):

\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -1 & -7 \ 0 & -48 \ 1 & -171 \ 2 & -376 \ 3 & -663 \ \hline \end{array} \]

Next, let’s examine how \(f(x)\) changes as \(x\) increases. A common approach to determining the type of function is to look at the differences between successive values of \(f(x)\).

1. Calculate the first differences (\(f(x+1) - f(x)\)):

   • For \(x = -1\) to \(0\): \(-48 - (-7) = -41\)
   • For \(x = 0\) to \(1\): \(-171 - (-48) = -123\)
   • For \(x = 1\) to \(2\): \(-376 - (-171) = -205\)
   • For \(x = 2\) to \(3\): \(-663 - (-376) = -287\)

   First differences: \(-41, -123, -205, -287\)

2. Next, calculate the second differences:

   • From \(-41\) to \(-123\): \(-123 - (-41) = -82\)
   • From \(-123\) to \(-205\): \(-205 - (-123) = -82\)
   • From \(-205\) to \(-287\): \(-287 - (-205) = -82\)

   Second differences: \(-82, -82, -82\)

Since the second differences are constant, we can conclude that \(f(x)\) is a quadratic function.

Thus, we can fill in the answer:

This function is a quadratic function because the second differences of \(f(x)\) are constant.

Answer 2

why is it a quadratic function

Answer 3

A function is classified as a quadratic function if it can be expressed in the standard form:

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

where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\). Quadratic functions have specific characteristics, including their graphs being parabolas.

To determine if the given function is quadratic, we evaluated the differences of the function values in the provided table. Let's recap the steps to emphasize why the function is quadratic:

1. First Differences: When we calculated the first differences (the differences between consecutive values of \(f(x)\)), we observed that the change in \(f(x)\) was not constant; it was varying since values became more spaced apart.

2. Second Differences: We then computed the second differences (the differences between consecutive first differences). For a quadratic function, the second differences are constant.

Here are the calculated differences again:

• First Differences:

  • From \(f(-1)\) to \(f(0)\): \(-41\)
  • From \(f(0)\) to \(f(1)\): \(-123\)
  • From \(f(1)\) to \(f(2)\): \(-205\)
  • From \(f(2)\) to \(f(3)\): \(-287\)

• Second Differences:

  • \(-82\) (from \(-41\) to \(-123\))
  • \(-82\) (from \(-123\) to \(-205\))
  • \(-82\) (from \(-205\) to \(-287\))

Since the second differences are constant (in this case, \(-82\)), this indicates that the relationship between \(x\) and \(f(x)\) is described by a quadratic equation.

Summary

The constant second differences in the computed values reflect the fundamental characteristic of quadratic functions. In summary, the function in the table is quadratic because it displays constant second differences, a key feature of all quadratic polynomials.