Determine whether the table of values represents a linear, quadratic, or exponential function.
x
−
7
−
6
−
5
−
4
f
(
x
)
166
124
88
58
x
f(x)
​

−7
166
​

−6
124
​

−5
88
​

−4
58
​

​

Quadratic

Linear

Exponential

None of these
​​​
Which statement justifies the type of function that it is?

As
x
x​ increases at a constant rate, the rate at which the values of
f
(
x
)
f(x)​ increase, or decrease, is constant.

As
x
x​ increases at a constant rate, the rate at which the values of
f
(
x
)
f(x)​ increase, or decrease, is multiplicative.

As
x
x​ increases at a constant rate, the second differences of the
f
(
x
)
f(x)​ values are the same.

None of these statements describes the characteristics of
f
(
x
)
f(x)​ as
x
x​ increases.
​

Let's analyze the table of values to determine the type of function it represents.

### Given values:
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-7 & 166 \\
-6 & 124 \\
-5 & 88 \\
-4 & 58 \\
\end{array}
\]

### Step 1: Determine if it's a linear function.
For a linear function, the first differences of \( f(x) \) should be constant. Let's calculate the first differences:

\[
\begin{array}{c|c}
\text{Change in } x & \text{Change in } f(x) \\
\hline
-7 \text{ to } -6 & 124 - 166 = -42 \\
-6 \text{ to } -5 & 88 - 124 = -36 \\
-5 \text{ to } -4 & 58 - 88 = -30 \\
\end{array}
\]

Since the first differences \(-42\), \(-36\), and \(-30\) are not constant, the function is **not linear**.

### Step 2: Determine if it's a quadratic function.
For a quadratic function, the second differences of \( f(x) \) should be constant. Let's calculate the second differences:

\[
\begin{array}{c|c}
\text{First Difference} & \text{Second Difference} \\
\hline
-42 \text{ to } -36 & -36 - (-42) = 6 \\
-36 \text{ to } -30 & -30 - (-36) = 6 \\
\end{array}
\]

Since the second differences \(6\) and \(6\) are constant, the function is **quadratic**.

### Conclusion:
The table of values represents a **quadratic function**.

### Justification:
The correct statement that justifies the type of function is:
**"As \( x \) increases at a constant rate, the second differences of the \( f(x) \) values are the same."**

Would you like more details or have any questions about this analysis?

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Here are 5 related questions:
1. How would you determine if a function is exponential using a table of values?
2. What are the general forms of linear, quadratic, and exponential functions?
3. Can a function be classified as both linear and quadratic?
4. What is the significance of second differences in identifying quadratic functions?
5. How can you verify the type of function if given a graph instead of a table?

**Tip:** For quadratic functions, always check the second differences when the first differences are not constant; this is a key indicator of quadratic behavior.

Learn how to determine whether a table of values represents a linear, quadratic, or exponential function. Analyze the relationship between x and f(x), understand first and second differences, and justify the function type based on mathematical principles.

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