1. **Problem:** Sketch the graph of the function $g(x) = 3x$. 2. **Formula and rules:** This is a linear function of the form $y = mx + b$ where $m=3$ and $b=0$. The graph is a straight line passing through the origin with slope 3. 3. **Table of values:** - For $x=-2$, $g(-2) = 3(-2) = -6$ - For $x=-1$, $g(-1) = 3(-1) = -3$ - For $x=0$, $g(0) = 0$ - For $x=1$, $g(1) = 3$ - For $x=2$, $g(2) = 6$ 4. **Graph shape:** The line passes through points $(-2,-6)$, $(-1,-3)$, $(0,0)$, $(1,3)$, and $(2,6)$. It rises steeply because the slope is 3. --- 1. **Problem:** Sketch the graph of the piecewise function $$C(x) = \begin{cases} 5, & \text{if } x=3 \\ x-2, & \text{if } x \neq 3 \end{cases}$$ 2. **Explanation:** For all $x$ except 3, the function behaves like $y = x-2$, a line with slope 1 and y-intercept -2. At $x=3$, the function value is explicitly 5, which creates a jump (a point not on the line). 3. **Table of values for $x-2$ part:** - $x=1$, $C(1) = 1-2 = -1$ - $x=2$, $C(2) = 0$ - $x=4$, $C(4) = 2$ 4. **Graph shape:** The graph is the line $y=x-2$ with a hole at $x=3$ and a filled point at $(3,5)$. --- 1. **Problem:** Sketch the graph of the function $E(x) = \lfloor x \rfloor + 5$ where $\lfloor x \rfloor$ is the floor function. 2. **Explanation:** The floor function $\lfloor x \rfloor$ returns the greatest integer less than or equal to $x$. Adding 5 shifts the step graph up by 5 units. 3. **Table of values:** - For $x=1.2$, $E(1.2) = \lfloor 1.2 \rfloor + 5 = 1 + 5 = 6$ - For $x=2.9$, $E(2.9) = 2 + 5 = 7$ - For $x=3$, $E(3) = 3 + 5 = 8$ 4. **Graph shape:** The graph is a step function starting at $y=5$ for $0 \leq x < 1$, then jumps to $y=6$ for $1 \leq x < 2$, and so on. --- **Final answers:** - $g(x) = 3x$ is a straight line through the origin with slope 3. - $C(x)$ is a line $y=x-2$ with a hole at $x=3$ and a filled point at $(3,5)$. - $E(x) = \lfloor x \rfloor + 5$ is a step function shifted up by 5.