https://file.aiquickdraw.com/mathbot/file/1727894273119hshj86gz.jpg

To estimate the slope of the tangent line at the given point (-2, 3) on the curve in the image, we can look at the tangent line drawn through that point.

The slope of a line is calculated using the formula:

\[
\text{slope} = \frac{\Delta y}{\Delta x}
\]

We need two points on the tangent line to calculate the slope. From the graph, one point is clearly labeled as (-2, 3). Looking at another clear point where the tangent line crosses the grid (it appears to be at the point (0, 5)) allows us to calculate the change in \(y\) and \(x\):

- Change in \(x\) (\(\Delta x\)): From -2 to 0 is a change of 2 units.
- Change in \(y\) (\(\Delta y\)): From 3 to 5 is a change of 2 units.

Therefore, the slope of the tangent line is:

\[
\text{slope} = \frac{5 - 3}{0 - (-2)} = \frac{2}{2} = 1
\]

So, the slope of the tangent line is approximately **1**.

Would you like further details on this explanation? Here are some related questions to explore:

1. How would the slope change if the curve were steeper?
2. Can you explain why the slope of a tangent line represents the derivative at that point?
3. How can we find the equation of the tangent line using this slope?
4. What is the significance of the slope being positive in this case?
5. How could you calculate the slope algebraically if the equation of the curve were given?

**Tip:** The slope of the tangent line at a point on a curve is the value of the derivative at that point.

Estimate the slope of the tangent line to the curve at the given point (x, y).

Learn how to estimate the slope of a tangent line to a curve at a given point using graphical analysis. This problem involves key calculus concepts such as derivatives, tangent lines, and the slope formula. Suitable for students in Grades 11-12.

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