To determine the equation that matches the graph, we need to identify the different parts and behaviors of the function shown. Here's an analysis of the given graph:

1. For $x < -2$: The function appears to be decreasing in a parabolic shape.
2. For $-2 \leq x < 1$: The function is a continuous curve that increases and then hits a specific point.
3. For $1 \leq x < 2$: The function is constant.
4. For $x \geq 2$: The function increases linearly.

Let's write the piecewise function:

Piecewise function components:

1. For $x < -2$: The function is a part of a parabola. $f(x) = (x + 2)^2$ This matches the shape we see for $x < -2$.

2. For $-2 \leq x < 1$: This part transitions smoothly from the parabolic section to the point (1,1). $f(x) = 1$ This represents the constant part of the graph for $-2 \leq x < 1$.

3. For $x = 1$: The function value is $f(1) = 2$.

4. For $1 < x < 2$: The function is again constant but at a different value. $f(x) = 3$

5. For $x \geq 2$: The function increases linearly. $f(x) = x + 1$ This represents the linear increase for $x \geq 2$.

Complete piecewise function:

\begin{cases} (x + 2)^2 & \text{if } x < -2 \\ 1 & \text{if } -2 \leq x < 1 \\ 2 & \text{if } x = 1 \\ 3 & \text{if } 1 < x < 2 \\ x + 1 & \text{if } x \geq 2 \end{cases}$$ ### Verification: - **For $$x < -2$$**, $$(x + 2)^2$$ is the equation of the parabola. - **For $$-2 \leq x < 1$$**, $$f(x) = 1$$ is a constant function. - **At $$x = 1$$**, $$f(x) = 2$$. - **For $$1 < x < 2$$**, $$f(x) = 3$$ is another constant function. - **For $$x \geq 2$$**, $$f(x) = x + 1$$ is a linear function. This matches the behavior and values shown in the graph. Would you like more details or have any questions? --- ### Questions for Further Exploration: 1. How can we verify the continuity of the function at each boundary point? 2. What are the intercepts of this piecewise function with the x-axis and y-axis? 3. How would you plot this piecewise function on a coordinate plane? 4. Can you find the domain and range of this piecewise function? 5. How does the function behave as $$x$$ approaches $$-2$$ from the left and the right? 6. How does the function behave as $$x$$ approaches 1 from the left and the right? 7. How does the function behave as $$x$$ approaches 2 from the left and the right? 8. How would the function change if the linear part for $$x \geq 2$$ was $$2x - 1$$ instead? ### Tip: When working with piecewise functions, ensure that you check the continuity and differentiability at the points where the pieces meet. This helps in understanding the overall behavior of the function.