For the quadratic equation, state the axis of symmetry, vertex, and any intercepts.
y = x2 − 4x − 5
axis of symmetry

vertex
(x, y) =

y-intercept
(x, y) =

x-intercept
(x, y) =

(smaller x-value)
x-intercept
(x, y) =

(larger x-value)

Graph the quadratic equation.

Let's break down the problem step by step for the quadratic equation \( y = x^2 - 4x - 5 \).

### 1. **Axis of Symmetry**
For a quadratic equation of the form \( y = ax^2 + bx + c \), the axis of symmetry is given by the formula:

\[
x = \frac{-b}{2a}
\]

Here, \( a = 1 \), \( b = -4 \), and \( c = -5 \). Substituting into the formula:

\[
x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2
\]

So, the axis of symmetry is:

\[
x = 2
\]

### 2. **Vertex**
The vertex is the point where the parabola changes direction. The x-coordinate of the vertex is the same as the axis of symmetry, which is \( x = 2 \). To find the y-coordinate of the vertex, substitute \( x = 2 \) into the original equation \( y = x^2 - 4x - 5 \):

\[
y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9
\]

So, the vertex is:

\[
(x, y) = (2, -9)
\]

### 3. **Y-Intercept**
The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = (0)^2 - 4(0) - 5 = -5
\]

So, the y-intercept is:

\[
(x, y) = (0, -5)
\]

### 4. **X-Intercepts**
The x-intercepts occur when \( y = 0 \). To find the x-intercepts, we solve \( x^2 - 4x - 5 = 0 \). Factor the quadratic equation:

\[
(x - 5)(x + 1) = 0
\]

Setting each factor to 0:

\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]

So, the x-intercepts are:

\[
(5, 0) \quad \text{(larger x-value)}
\]
\[
(-1, 0) \quad \text{(smaller x-value)}
\]

### 5. **Graph of the Quadratic Equation**
To graph the equation \( y = x^2 - 4x - 5 \), we use the vertex, axis of symmetry, and intercepts found above. The parabola will open upwards because the coefficient of \( x^2 \) is positive.

- The vertex is at \( (2, -9) \).
- The axis of symmetry is the vertical line \( x = 2 \).
- The y-intercept is \( (0, -5) \).
- The x-intercepts are \( (-1, 0) \) and \( (5, 0) \).

Would you like me to graph this for you as well?

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#### 5 Related Questions:
1. How do you derive the formula for the axis of symmetry?
2. What effect does changing the coefficient of \( x^2 \) have on the graph?
3. Can we graph quadratic equations that don’t factor easily?
4. What happens to the vertex if you change the constant in a quadratic equation?
5. How do you use completing the square to find the vertex?

#### Tip:
The vertex form of a quadratic equation, \( y = a(x-h)^2 + k \), makes it easy to identify the vertex \((h, k)\). Converting to this form helps in graphing.

For the quadratic equation, state the axis of symmetry, vertex, and any intercepts.
y = x^2 − 4x − 5
axis of symmetry
vertex (x, y) =
y-intercept (x, y) =
x-intercept (x, y) =
(smaller x-value)
x-intercept (x, y) =
(larger x-value)
Graph the quadratic equation.

Explore how to find the axis of symmetry, vertex, and intercepts for the quadratic equation y = x^2 - 4x - 5. This step-by-step guide walks through algebraic techniques, including factoring and symmetry analysis. Suitable for students in Grades 8-10.

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