Question

how do I graph y\ =\ \left\{30x\ if\ 0\ <\ x\ \le\ 3\ 75\ if\ 3\ <\ x\ \le\ 7\ 20x\ if\ x\ >\ 7\right\}

Answer 1

To graph the piecewise function

\[
y =
\begin{cases}
30x & \text{if } 0 < x \leq 3 \\
75 & \text{if } 3 < x \leq 7 \\
20x & \text{if } x > 7
\end{cases}
\]

follow the steps below:

### Step 1: Identify intervals and corresponding equations

1. **For \(0 < x \leq 3\)**: The function is linear, given by the equation \(y = 30x\).
2. **For \(3 < x \leq 7\)**: The function is constant, given by \(y = 75\).
3. **For \(x > 7\)**: The function is again linear, given by the equation \(y = 20x\).

### Step 2: Calculate key points

- For \(y = 30x\):
- At \(x = 1\), \(y = 30(1) = 30\)
- At \(x = 2\), \(y = 30(2) = 60\)
- At \(x = 3\), \(y = 30(3) = 90\)

- For \(y = 75\):
- At \(x = 4\), \(y = 75\)
- At \(x = 5\), \(y = 75\)
- At \(x = 6\), \(y = 75\)
- At \(x = 7\), \(y = 75\)

- For \(y = 20x\):
- At \(x = 8\), \(y = 20(8) = 160\)
- At \(x = 9\), \(y = 20(9) = 180\)

### Step 3: Plot the points

1. **For the segment \(y = 30x\)** from \(0 < x \leq 3\):
- Plot the points \((1, 30)\), \((2, 60)\), and \((3, 90)\).
- Draw a line segment from \((0, 0)\) to \((3, 90)\) but **do not include** the point at \(x = 0\) (open circle) and include at \(x = 3\) (closed circle).

2. **For the segment \(y = 75\)** from \(3 < x \leq 7\):
- Plot the points \((4, 75)\), \((5, 75)\), \((6, 75)\), and \((7, 75)\).
- Draw a horizontal line at \(y = 75\) from \(x = 3\) (open circle) to \(x = 7\) (closed circle).

3. **For the segment \(y = 20x\)** for \(x > 7\):
- Plot the points \((8, 160)\) and \((9, 180)\).
- Draw a line starting from \((7, 140)\) onwards to the right (open circle at \(x = 7\) and line continues to the right).

### Step 4: Finish the graph

- Label your axes (x and y).
- Clearly mark the open and closed circles where appropriate.
- You should end up with three distinct pieces: a rising line from 0 to 3, a horizontal line from 3 to 7, and another rising line starting at 7 and continuing upwards.

### Final Note
Ensure that you draw your graph carefully, maintaining the distinct segments and their respective behaviors as described in the piecewise function.