Given the description of the curve and the points $P(-3.2, -1.4)$, $Q(1.2, 0.8)$, and $R(4.4, 1.4)$, let's analyze the behavior of the tangent slopes at these points. To match the slopes to the correct points, we need to determine the points on the graph where the tangent has a specific slope, based on the shape and direction of the curve.

1. Slope of 0 (Horizontal Tangent Line)

• A slope of zero corresponds to a horizontal tangent. This occurs when the curve changes direction, i.e., at local minimum or maximum points.
• According to the description, the curve changes direction twice:
  • At $(-2, -2)$ in the second quadrant (likely a local minimum).
  • At $(2.6, 3.6)$ in the first quadrant (likely a local maximum).

Thus, one of these points must have a horizontal tangent. Based on the points given in the problem, the point closest to $(2.6, 3.6)$ is likely near point Q because it lies in the increasing section before the direction change. Therefore, the slope of 0 occurs at Q.

2. Slope of 1 (Positive Slope)

• A slope of 1 corresponds to a diagonal line that increases at a 45° angle.
• This slope is typically seen when the curve is increasing steeply but not too sharply.
• The curve passes through $R(4.4, 1.4)$ after a direction change, and based on the description, the slope becomes positive and increases.

Thus, the slope of 1 occurs at R.

3. Slope of -1 (Negative Slope)

• A slope of -1 corresponds to a diagonal line that decreases at a 45° angle.
• This slope is typically seen when the curve is decreasing steadily.
• The curve passes through $P(-3.2, -1.4)$ when it is decreasing, and based on the description, this is before it reaches its lowest point at $(-2, -2)$.

Thus, the slope of -1 occurs at P.

Final Answers:

• (a) Slope 0: Point Q
• (b) Slope 1: Point R
• (c) Slope -1: Point P

Would you like more details on how to interpret tangent slopes, or do you have any questions?

Here are some related questions to expand your understanding:

1. How do we determine the slope of a tangent line at a specific point on a curve?
2. What is the geometric significance of a slope of 0 for a function?
3. How does a local maximum or minimum affect the tangent slope of a function?
4. Can a curve have multiple points with the same slope?
5. What other methods can be used to estimate tangent slopes without a graph?

Tip: To quickly estimate slopes visually, look at the steepness of the curve at different points; flatter sections correspond to slopes near 0, while steeper sections show larger positive or negative slopes.