Graph the relationship \( h=10 t-t^{2} \).

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!