1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} 1 + x^2 & \text{if } x < 1 \\ 9x - 7 & \text{if } x \geq 1 \end{cases}$$ We want to understand the behavior and key points of this function. 2. **Analyze the first piece:** For $x < 1$, the function is $f(x) = 1 + x^2$. - This is a parabola opening upwards with vertex at $(0,1)$. - Since $x^2 \geq 0$, the minimum value in this region is at $x=0$, $f(0) = 1$. 3. **Analyze the second piece:** For $x \geq 1$, the function is $f(x) = 9x - 7$. - This is a linear function with slope 9 and y-intercept $-7$. - At $x=1$, $f(1) = 9(1) - 7 = 2$. 4. **Check continuity at $x=1$:** - Left limit: $\lim_{x \to 1^-} f(x) = 1 + 1^2 = 2$ - Right limit: $f(1) = 2$ - Since both are equal, the function is continuous at $x=1$. 5. **Summary of graph shape:** - For $x < 1$, the graph is a parabola starting from $x=-12$ up to just before $x=1$, with values $f(x) = 1 + x^2$. - At $x=1$, the function value is 2. - For $x \geq 1$, the graph is a straight line with slope 9 starting at $(1,2)$. 6. **Final answer:** The piecewise function is continuous at $x=1$ and consists of an upward parabola for $x<1$ and a steep increasing line for $x \geq 1$.