Question

Isabel is tracking a plant's height for her science class. When she begins tracking, the plant is 3 centimeters tall. The plant then grows 3 centimeters per week. Let x be the time (in weeks) since Isabel started tracking the plant's height, and let y be its height (in centimeters).
(a)Graph the line that shows the relationship between x and y.

Answer 1

To graph the relationship between time \( x \) (in weeks) and the plant's height \( y \) (in centimeters), we start by establishing the equation of the line based on the information given.

The initial height of the plant at \( x = 0 \) is 3 centimeters, and it grows at a rate of 3 centimeters per week. Therefore, we can express the height \( y \) as a function of time \( x \):

\[
y = 3 + 3x
\]

Where:
- The \( 3 \) represents the initial height of the plant.
- The \( 3x \) term represents the growth of the plant over time.

### Steps to create the graph:

1. **Plot the y-intercept**: When \( x = 0 \), \( y = 3 \). So, the point (0, 3) lies on the graph.
2. **Calculate another point**:
- For \( x = 1 \):
\[
y = 3 + 3(1) = 6 \quad \text{(Point (1, 6))}
\]
- For \( x = 2 \):
\[
y = 3 + 3(2) = 9 \quad \text{(Point (2, 9))}
\]
- For \( x = 3 \):
\[
y = 3 + 3(3) = 12 \quad \text{(Point (3, 12))}
\]

3. **Draw the line**: After plotting the points (0, 3), (1, 6), (2, 9), and (3, 12), you can draw a straight line through these points. This line will have a positive slope, indicating that the height of the plant increases over time.

### Graph Representation:

- The x-axis (horizontal) represents time \( x \) in weeks.
- The y-axis (vertical) represents the height \( y \) in centimeters.
- The line will start at the point (0, 3) and slope upwards to the right.

### Summary of Points:
- (0, 3) — the initial height
- (1, 6) — height after 1 week
- (2, 9) — height after 2 weeks
- (3, 12) — height after 3 weeks

Feel free to plot these points on graph paper or use graphing software to visualize the linear relationship!