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Answer
To graph the equation \( h = 10t - t^{2} \):
1. **Vertex**: At \( t = 5 \), \( h = 25 \).
2. **Intercepts**: \( t = 0 \) and \( t = 10 \) on the \( t \)-axis; \( h = 0 \) on the \( h \)-axis.
3. **Axis of Symmetry**: \( t = 5 \).
4. **Shape**: A downward-opening parabola.
Plot these points and sketch the parabola accordingly.
Solution
Certainly! Let's explore the graph of the quadratic relationship \( h = 10t - t^{2} \).
### 1. **Understanding the Equation**
The equation \( h = 10t - t^{2} \) is a quadratic equation in the form:
\[
h(t) = -t^{2} + 10t
\]
This represents a parabola that opens **downwards** because the coefficient of \( t^{2} \) is negative.
### 2. **Key Features of the Graph**
- **Vertex**: The highest point on the parabola.
- **Axis of Symmetry**: A vertical line that passes through the vertex.
- **Intercepts**: Points where the graph crosses the axes.
#### a. **Finding the Vertex**
The vertex of a parabola given by \( h(t) = at^{2} + bt + c \) can be found using:
\[
t = -\frac{b}{2a}
\]
Here, \( a = -1 \) and \( b = 10 \):
\[
t = -\frac{10}{2(-1)} = 5
\]
Now, find \( h \) when \( t = 5 \):
\[
h(5) = 10(5) - (5)^{2} = 50 - 25 = 25
\]
**Vertex**: \( (5, 25) \)
#### b. **Axis of Symmetry**
The axis of symmetry is the vertical line that passes through the vertex:
\[
t = 5
\]
#### c. **Finding the Intercepts**
- **T-intercepts (where \( h = 0 \))**:
\[
10t - t^{2} = 0 \\
t(10 - t) = 0 \\
t = 0 \quad \text{or} \quad t = 10
\]
So, the graph intersects the \( t \)-axis at \( t = 0 \) and \( t = 10 \).
- **H-intercept (where \( t = 0 \))**:
\[
h(0) = 10(0) - (0)^{2} = 0
\]
So, the graph intersects the \( h \)-axis at \( h = 0 \).
### 3. **Sketching the Graph**
Here's a step-by-step guide to sketching the graph:
1. **Plot the Vertex**: \( (5, 25) \)
2. **Plot the Intercepts**:
- \( t \)-intercepts at \( (0, 0) \) and \( (10, 0) \)
- \( h \)-intercept at \( (0, 0) \) (same as one \( t \)-intercept)
3. **Draw the Axis of Symmetry**: A dashed vertical line at \( t = 5 \)
4. **Shape of the Parabola**: Since the coefficient of \( t^{2} \) is negative, the parabola opens downward.
5. **Plot Additional Points** (optional for accuracy):
- For \( t = 2 \): \( h(2) = 20 - 4 = 16 \) → \( (2, 16) \)
- For \( t = 8 \): \( h(8) = 80 - 64 = 16 \) → \( (8, 16) \)
### 4. **Graph Visualization**
While I can't display images directly, here's a Python script using `matplotlib` that you can run to visualize the graph:
```python
import matplotlib.pyplot as plt
import numpy as np
# Define the function
def h(t):
return 10*t - t**2
# Generate t values
t = np.linspace(-2, 12, 400)
h_values = h(t)
# Plot the function
plt.figure(figsize=(8,6))
plt.plot(t, h_values, label='h = 10t - t²', color='blue')
# Plot the vertex
plt.plot(5, 25, 'ro', label='Vertex (5, 25)')
# Plot intercepts
plt.plot(0, 0, 'go', label='Intercept (0,0)')
plt.plot(10, 0, 'go', label='Intercept (10,0)')
# Draw axis of symmetry
plt.axvline(x=5, color='gray', linestyle='--', label='Axis of Symmetry t=5')
# Set labels and title
plt.xlabel('t')
plt.ylabel('h(t)')
plt.title('Graph of h = 10t - t²')
plt.legend()
plt.grid(True)
plt.xlim(-1, 11)
plt.ylim(-5, 30)
plt.show()
```
**Instructions to Run the Script:**
1. Ensure you have Python installed on your computer.
2. Install `matplotlib` and `numpy` if you haven't already by running:
```
pip install matplotlib numpy
```
3. Copy and paste the script into a Python file (e.g., `plot_h.py`) or an interactive environment like Jupyter Notebook.
4. Run the script to see the graph.
### 5. **Description of the Graph**
- The parabola starts at the origin \( (0, 0) \), rises to the vertex at \( (5, 25) \), and then falls back down to \( (10, 0) \).
- The symmetry axis at \( t = 5 \) divides the parabola into two mirror images.
- The maximum value of \( h \) is 25 when \( t = 5 \).
### 6. **Applications**
Such quadratic relationships often model projectile motion where:
- \( h(t) \) represents the height of an object at time \( t \).
- The object is projected upwards with an initial velocity (here, 10 units/time).
- The \( -t^{2} \) term accounts for the acceleration due to gravity.
### 7. **Final Thoughts**
Understanding the key features of quadratic equations allows you to graph them accurately and interpret their real-world applications effectively. Feel free to run the provided Python script to visualize the graph and experiment with different values to deepen your understanding!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To graph the relationship \( h = 10t - t^{2} \), start by recognizing that this is a quadratic equation that opens downward (since the coefficient of \( t^2 \) is negative). The graph will be a parabola. To find the vertex, use the formula \( t = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = 10 \), which gives us \( t = 5 \). This means the maximum height occurs at \( t = 5 \), substituting back gives you \( h = 10(5) - 5^2 = 25 \). Next, evaluate the function at key points around the vertex, like \( t = 0 \) (where \( h = 0 \)) and \( t = 10 \) (also \( h = 0 \)). Plot these points and sketch the parabola, noting that it intersects the \( h \)-axis at \( (0,0) \) and \( (10,0) \), with the highest point at \( (5,25) \) creating a beautiful arch.
