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The vertex of a parabola represents its highest or lowest point, depending on whether the parabola opens upwards or downwards. For the quadratic function given as \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex is found using the formula \(x = -\frac{b}{2a}\). This formula helps us pinpoint the exact horizontal location of the vertex. For our specific function \(f(x) = x^2 - 4x - 5\), we substitute \(a = 1\) and \(b = -4\) to get: \[ x = -\frac{-4}{2 \times 1} = 2 \]
To find the y-coordinate, we substitute \(x = 2\) back into the function and solve for \(y\): \[ y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9 \] This means the vertex is located at the point \( (2, -9) \).
The vertex is a critical point that helps in sketching the parabola correctly.

The axis of symmetry of a parabola is a vertical line that passes through its vertex, dividing the parabola into two mirror-image halves. This axis of symmetry has a simple equation derived directly from the x-coordinate of the vertex. Using the vertex we found earlier \( (2, -9)\), the axis of symmetry for our given quadratic function is: \[ x = 2 \]
This vertical line helps in ensuring the parabola is symmetric about this axis when graphing. Knowing the axis of symmetry can provide a clear guideline on how to plot additional points on the parabola.

Intercepts are points where the graph crosses the axes. Identifying these helps in accurately sketching the parabola.

**Finding the Y-Intercept:**
The y-intercept is found by setting \( x = 0 \) and solving for \( y \). For \(f(x) = x^2 - 4x - 5\), substituting \(x = 0\) gives us: \[ f(0) = 0^2 - 4(0) - 5 = -5 \] Therefore, the y-intercept is the point \( (0, -5) \).

**Finding the X-Intercepts:**
X-intercepts are found by setting the function equal to zero and solving for \( x \). This involves solving the equation: \[ x^2 - 4x - 5 = 0 \] Factoring the quadratic equation, we get: \[ (x - 5)(x + 1) = 0 \] Solving for \( x \), we find: \[ x = 5 \] and \[ x = -1 \] So, the x-intercepts are \( (5, 0) \) and \( (-1, 0) \).
Understanding intercepts is vital in sketching the precise shape of the parabola.

Factoring is a method used to solve quadratic equations and find the x-intercepts of a parabola. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). To factor the quadratic, you need to express it as a product of two binomials.

For our given quadratic equation \(x^2 - 4x - 5 = 0\), we look for two numbers that multiply to \(-5\) and add up to \(-4\). These numbers are \(-5\) and \(+1\): \[ x^2 - 4x - 5 = (x - 5)(x + 1) \]
Once factored, we set each binomial to zero to find the solutions for \ x \):
\( x - 5 = 0 \text{ or } \( x + 1 = 0 \).
Solving these gives us: \[ x = 5 \] and \[ x = -1 \]
Thus, the x-intercepts are \(5 \) and \(-1 \). Factoring is a powerful tool that simplifies solving quadratic equations and understanding the points where the parabola crosses the x-axis.