Let me analyze the content in the image and solve the problem for you. One moment, please!### Problem Breakdown: From the graph and the problem:

1. Horizontal Asymptote (asymptota pozioma): This corresponds to the value $y = c$ the function approaches as $x \to \pm\infty$.

2. Vertical Asymptote (asymptota pionowa): This is the value $x = k$ where the function is undefined, typically causing the function to diverge to $\pm \infty$.

3. Tangent Slope at $x = -2$: Using $a = f'(-2)$, calculate the derivative of the function at $x = -2$.

4. Equation of the Tangent Line: The tangent line at $A(-2, 2)$ is of the form $y = mx + b$, where $m$ is the slope from $f'(-2)$, and $b$ can be determined by substituting the point $(-2, 2)$.

Observations from the Graph:

1. Horizontal Asymptote: The function approaches $y = 0$ as $x \to \pm \infty$.

   Answer: $y = 0$.

2. Vertical Asymptote: The graph becomes undefined at $x = -1$, which is the vertical asymptote.

   Answer: $x = -1$.

3. Tangent Slope at $x = -2$: From the graph, the tangent line is given, and its slope is equal to the derivative $f'(-2)$. The slope is visually approximated as $a = 1$.

   Answer: $a = 1$.

4. Equation of the Tangent Line: The tangent line passes through the point $(-2, 2)$ with a slope of $1$. Using the point-slope formula: $y - y_1 = m(x - x_1)$ Substituting $m = 1$, $x_1 = -2$, and $y_1 = 2$: $y - 2 = 1(x + 2)$ $y = x + 4$

   Answer: $y = x + 4$.

Final Answers:

1. Horizontal asymptote: $y = 0$.
2. Vertical asymptote: $x = -1$.
3. Tangent slope at $x = -2$: $a = 1$.
4. Equation of the tangent line: $y = x + 4$.

Would you like detailed calculations or explanations for any of the steps?

Expanding Questions:

1. How can we formally calculate the derivative $f'(x)$ from the graph?
2. Why does the horizontal asymptote represent the long-term behavior of the function?
3. How do vertical asymptotes relate to the domain of the function?
4. What other methods can be used to find the tangent line equation if the slope isn't obvious?
5. How do the tangent line and the function relate geometrically at a given point?

Tip:

When working with graphs, always cross-check the slope of the tangent visually and algebraically if possible for higher accuracy.