1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} -x + 3, & x > 0 \\ 2x + 3, & x \leq 0 \end{cases}$$ We want to understand the shape of its graph. 2. **Recall the formula and rules:** Each piece is a linear function of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Analyze each piece:** - For $x > 0$, the function is $f(x) = -x + 3$. This line has slope $-1$ and y-intercept $3$. - For $x \leq 0$, the function is $f(x) = 2x + 3$. This line has slope $2$ and y-intercept $3$. 4. **Graph shape description:** - For $x > 0$, the graph is a line descending from the point $(0,3)$ with slope $-1$. - For $x \leq 0$, the graph is a line ascending from the point $(0,3)$ with slope $2$. 5. **Check continuity at $x=0$:** Both pieces meet at $x=0$ with $f(0) = 3$, so the graph is continuous there. **Final answer:** The graph consists of two linear pieces joined at $(0,3)$, with the left piece rising steeply (slope 2) and the right piece falling gently (slope -1).