Given the data points:

• $x = -2, -1, 0, 1, 2$
• $y = -5, 0, 2, 4, 5$

Let's go through each part:

(a) Plot the points $(x, y)$ in a coordinate plane.

The coordinates given are:

• $(-2, -5)$
• $(-1, 0)$
• $(0, 2)$
• $(1, 4)$
• $(2, 5)$

These can be plotted in a standard 2D Cartesian plane.

(b) Find the slope of the line through the points.

To find the slope $m$ of a line through two points, we use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Let's compute the slope between two points, say $(-2, -5)$ and $(2, 5)$: $m = \frac{5 - (-5)}{2 - (-2)} = \frac{5 + 5}{2 + 2} = \frac{10}{4} = 2.5$

Thus, the slope is $m = 2.5$.

(c) Find the equation of the line.

Using the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ We can use the point $(0, 2)$ and the slope $m = 2.5$: $y - 2 = 2.5(x - 0)$ $y = 2.5x + 2$

Thus, the equation of the line is: $y = 2.5x + 2$

(d) Use the equation of the line to predict $y$ when $x = 3$.

Substitute $x = 3$ into the equation: $y = 2.5(3) + 2 = 7.5 + 2 = 9.5$ Thus, when $x = 3$, $y = 9.5$.

(e) Use the equation of the line to predict $y$ when $x = -3$.

Substitute $x = -3$ into the equation: $y = 2.5(-3) + 2 = -7.5 + 2 = -5.5$ Thus, when $x = -3$, $y = -5.5$.

(f) Is the prediction reasonable when $x = 3$?

Yes, the prediction is reasonable because the data points seem to follow a linear trend. Based on the slope and the pattern of change between values of $x$ and $y$, the predicted value at $x = 3$ falls within the expected range for this linear relationship.

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Would you like more details on any of these parts?

Here are five follow-up questions for you to explore:

1. What does the slope of 2.5 represent in this context?
2. How would the equation change if the slope were negative?
3. How can you verify if the equation fits all the data points?
4. What happens to the line's behavior as $x$ approaches very large or very small values?
5. Can you solve for $x$ if a specific value of $y$ is given?

Tip: Always check that your linear model fits the data well by verifying it against all provided points.