Question

use the graph to fill in an answer to the sentence below and find an explicit formula that matches the gra
the graph above can best be said to represent
options: an arithmetic sequence, a geometric sequence, a linear function, an exponential function

Explanation:

Step1: Analyze the differences between the options

First, let's recall the characteristics of each option:

• Arithmetic sequence: Has a common difference between consecutive terms (linear relationship when graphed as discrete points).
• Geometric sequence: Has a common ratio between consecutive terms (exponential relationship when graphed as discrete points).
• Linear function: Has a constant slope (straight line when graphed).
• Exponential function: Has a constant ratio of change (curved line, often with rapid growth or decay).

Step2: Analyze the given points

The points given are \((-3, -1)\), \((-2, -3)\), \((-1, -9)\). Let's check the ratios between consecutive \(y\)-values:

• From \(x = -3\) to \(x = -2\): \(\frac{-3}{-1} = 3\)
• From \(x = -2\) to \(x = -1\): \(\frac{-9}{-3} = 3\)

There is a common ratio of \(3\), which is characteristic of a geometric sequence (or an exponential function, but since the points are discrete and the ratio is constant, a geometric sequence is more appropriate here as it's a sequence of discrete points with a common ratio). The graph is a curve, not a straight line, so it's not a linear function or arithmetic sequence. An exponential function is usually for continuous data, while a geometric sequence is for discrete data with a common ratio.

Answer:

a geometric sequence