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The vertex of a parabola is one of its most important features. It represents the highest or lowest point of the parabola depending on its direction. In our quadratic function, which is a downward-opening parabola, the vertex is the maximum point.
To find the vertex of the parabola given by the equation \( f(x) = -x^2 + 20x + 5 \), we use the formula for the vertex's x-coordinate: \( x = -\frac{b}{2a} \).

• Here, \( a = -1 \) and \( b = 20 \), which gives us \( x = -\frac{20}{2(-1)} = 10 \).

Substitute \( x = 10 \) back into the function to find the y-coordinate. \( f(10) = 5 + 20 \times 10 - 10^2 = 105 \). The vertex is therefore at the point \((10, 105)\). This vertex tells us where the parabola reaches its peak as it opens downwards.

The x-intercepts of a quadratic function are the points where the parabola crosses the x-axis. These points are found by setting the function equal to zero. For our function \( f(x) = 5 + 20x - x^2 \), set \( f(x) = 0 \): \[ 5 + 20x - x^2 = 0 \].
The rearrangement leads to \[ x^2 - 20x - 5 = 0 \]. We solve using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -1, b = 20, c = 5 \).

Calculate the discriminant: \( 20^2 - 4(-1)(5) = 420 \). Solve for x:

• \( x = \frac{20 \pm \sqrt{420}}{2} \).

Approximating \( \sqrt{420} \approx 20.5 \) gives the x-intercepts: approximately \( x \approx 0.25 \) and \( x \approx 20.25 \). These are the points \((0.25, 0)\) and \((20.25, 0)\).

The y-intercept of a quadratic function is the point where it crosses the y-axis. To find this, we set \( x = 0 \). For our function \( f(x) = 5 + 20x - x^2 \), this gives:

• \( f(0) = 5 + 20 \times 0 - 0^2 = 5 \).

Therefore, the y-intercept is the point \((0, 5)\).
This point is particularly useful as it quickly helps in plotting the initial structure of the parabola on a graph. It is the starting point for understanding the graph's behavior within the graph's initial range.

The quadratic formula is an essential tool for solving quadratic equations and finding the x-intercepts. It is especially useful when the equation does not factor easily. The formula is given as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In our equation \( 5 + 20x - x^2 = 0 \), the coefficients \( a = -1, b = 20, c = 5 \) determine the expression under the square root, known as the discriminant. Calculating the discriminant \( b^2 - 4ac \) gives us a value of 420, indicating we will have two real solutions.

These solutions are the x-intercepts, and finding them precisely involves computing:

• \( x = \frac{20 \pm \sqrt{420}}{2} \).

This method ensures we accurately determine where the parabola crosses the x-axis, providing critical information for correctly sketching the parabola.