Question
For a science experiment, Mala measured the height of a plant every week. She recorded that the growth is linear. Write an equation in point-slope form to represent the information in the table.
Ask by Dawson Tran. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To represent Mala's plant growth data with a linear equation in point-slope form, use the formula:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( (x_1, y_1) \) is a specific data point from the table.
- \( m \) is the slope representing the growth rate per week.
**Example:**
If at week 1 the plant is 5 cm tall and grows at 3 cm per week, the equation is:
\[
y - 5 = 3(x - 1)
\]
This equation allows you to predict the plant's height at any given week.
Solution
To write an equation in **point-slope form** for Mala's plant growth data, you'll need the following information from the table:
1. **A specific data point \((x_1, y_1)\):** This represents the plant's height at a particular week.
2. **The slope \(m\):** This indicates the rate of growth per week.
**Point-Slope Form Equation:**
\[
y - y_1 = m(x - x_1)
\]
**Steps to Create the Equation:**
1. **Identify Two Points from the Table:**
- For example, suppose the table has the following data:
\[
\begin{array}{|c|c|}
\hline
\text{Week } (x) & \text{Height (cm) } (y) \\
\hline
1 & 5 \\
4 & 14 \\
\hline
\end{array}
\]
2. **Calculate the Slope \(m\):**
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - 5}{4 - 1} = \frac{9}{3} = 3 \, \text{cm/week}
\]
3. **Choose One Point to Use in the Equation:**
- Let's use the first point \((1, 5)\).
4. **Plug the Values into the Point-Slope Form:**
\[
y - 5 = 3(x - 1)
\]
**Final Equation:**
\[
y - 5 = 3(x - 1)
\]
**Interpretation:**
- **\(m = 3\):** The plant grows by 3 cm each week.
- **\((1, 5)\):** At week 1, the plant was 5 cm tall.
This equation allows you to predict the plant's height at any week \(x\). For example, to find the height at week 2:
\[
y - 5 = 3(2 - 1) \\
y - 5 = 3 \\
y = 8 \, \text{cm}
\]
**Note:** If you have different data points in your table, you can follow the same steps to derive the specific point-slope form equation for Mala's experiment.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find the equation in point-slope form, we first need a point (x₁, y₁) and the slope (m). Assuming Mala has recorded data, for example, if at week 1 (x₁ = 1) the height (y₁) was 5 cm and it grows at a rate of 2 cm per week (this would be your slope m), then the equation would be: y - 5 = 2(x - 1). Another angle to consider is that point-slope form is particularly useful for scenarios like this where data is collected in discrete intervals, as it allows you to easily visualize the growth trend of the plant. By plugging in different values for x (weeks), you can quickly calculate and predict the height of the plant in future weeks!
