Question

select the correct answer.

which statement best describes the functions represented here?

table 1:
\\(\

$$\begin{array}{|c|c|} \\hline x & p(x) \\\\ \\hline 0 & 30 \\\\ \\hline 3 & 60 \\\\ \\hline 6 & 120 \\\\ \\hline 9 & 240 \\\\ \\hline \\end{array}$$

\\)

table 2:
\\(\

$$\begin{array}{|c|c|} \\hline x & q(x) \\\\ \\hline 0 & 20 \\\\ \\hline 3 & 110 \\\\ \\hline 6 & 380 \\\\ \\hline 9 & 830 \\\\ \\hline \\end{array}$$

\\)

• only function \\(p\\) is nonlinear.
• only function \\(q\\) is nonlinear.
• neither function is nonlinear.
• both functions are nonlinear.

Explanation:

🆕 New Concept Discovered: Linear vs. Nonlinear Functions
How to tell if a table shows a straight line

Step 1: Check the rate of change for function \( p(x) \)

A function is linear if it has a constant rate of change (slope). We calculate the change in \( p(x) \) divided by the change in \( x \) between consecutive rows:

From \( x = 0 \) to \( x = 3 \):

$$ \frac{60 - 30}{3 - 0} = \frac{30}{3} = 10 $$

From \( x = 3 \) to \( x = 6 \):

$$ \frac{120 - 60}{6 - 3} = \frac{60}{3} = 20 $$

Since the rate of change is not constant (\( 10
eq 20 \)), function \( p \) is nonlinear.

Step 2: Check the rate of change for function \( q(x) \)

We calculate the rate of change between consecutive rows for \( q(x) \):

From \( x = 0 \) to \( x = 3 \):

$$ \frac{110 - 20}{3 - 0} = \frac{90}{3} = 30 $$

From \( x = 3 \) to \( x = 6 \):

$$ \frac{380 - 110}{6 - 3} = \frac{270}{3} = 90 $$

Since the rate of change is not constant (\( 30
eq 90 \)), function \( q \) is also nonlinear.

Step 3: Compare results with the given options

Both function \( p \) and function \( q \) have changing rates of change, which means neither graph is a straight line. Therefore, both functions are nonlinear.

Answer:

Both functions are nonlinear.