To graph the equation $y = x^2 - 4x - 5$ and determine the roots, we will follow these steps:

1. Rewrite the equation: $y = x^2 - 4x - 5$.

2. Identify the key features:

   • The equation is a parabola (since it's quadratic) that opens upward (positive coefficient of $x^2$).
   • The vertex can be found using the formula for the x-coordinate of the vertex: $x = -\frac{b}{2a}$, where $a = 1$, $b = -4$, and $c = -5$.
   • The roots are the solutions to $x^2 - 4x - 5 = 0$, found using factoring, completing the square, or the quadratic formula.

3. Plot 5 points: The vertex, roots, and two additional points.

4. Determine roots using either the graph or direct computation.

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Steps:

Step 1: Vertex Calculation

$x = -\frac{-4}{2(1)} = 2.$ To find the y-coordinate of the vertex: $y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9.$ So, the vertex is $(2, -9)$.

Step 2: Factorize for Roots

Solve $x^2 - 4x - 5 = 0$: $x^2 - 4x - 5 = (x - 5)(x + 1) = 0.$ The roots are $x = 5$ and $x = -1$.

Step 3: Additional Points

Choose values of $x$ around the vertex ($x = 2$) to plot:

• At $x = 0$, $y = (0)^2 - 4(0) - 5 = -5$.
• At $x = 3$, $y = (3)^2 - 4(3) - 5 = 9 - 12 - 5 = -8$.

Plot Points:

1. Roots: $(-1, 0)$, $(5, 0)$
2. Vertex: $(2, -9)$
3. Additional points: $(0, -5)$, $(3, -8)$.

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Let's now graph the equation. numpy as np import matplotlib.pyplot as plt

Define the function y = x^2 - 4x - 5

def quadratic_function(x): return x**2 - 4*x - 5

Generate x values for the curve

x_values = np.linspace(-2, 6, 500) y_values = quadratic_function(x_values)

Points to plot

points = { "Roots": [(-1, 0), (5, 0)], "Vertex": [(2, -9)], "Additional Points": [(0, -5), (3, -8)] }

Plot the parabola

plt.figure(figsize=(8, 6)) plt.plot(x_values, y_values, label="y = x^2 - 4x - 5", color="blue")

Mark key points

for label, pts in points.items(): x_pts, y_pts = zip(*pts) plt.scatter(x_pts, y_pts, label=label, s=80)

Add labels and grid

plt.axhline(0, color='black',linewidth=0.7) plt.axvline(0, color='black',linewidth=0.7) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.title("Graph of y = x^2 - 4x - 5") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.show()The graph above displays the parabola $y = x^2 - 4x - 5$. Key points are:

• Roots: $(-1, 0)$ and $(5, 0)$.
• Vertex: $(2, -9)$.
• Additional Points: $(0, -5)$ and $(3, -8)$.

The roots of the equation $x^2 - 4x - 5 = 0$ are $x = -1$ and $x = 5$, confirmed by the graph and factorization.

Would you like further details or explanations about any part of this process?

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Related Questions:

1. How do you find the vertex of any quadratic function?
2. What is the significance of the roots in a quadratic equation?
3. How does the value of $a$ in $y = ax^2 + bx + c$ affect the shape of the parabola?
4. Can you derive the quadratic formula from completing the square?
5. How would the graph change if the equation were $y = -x^2 - 4x - 5$?

Tip: Always check symmetry in a parabola; it simplifies graphing and point selection.