Sketch the graph of the quadratic function.
f(x) = x2 − 20x − 800
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Indicate the coordinates of the vertex, the y-intercept and the x-intercepts (if any). (If an answer does not exist, enter DNE. If there are multiple x-intercepts, enter your answers as a comma-separated list.)
vertex	(x, y)	=

y-intercept	y	=
x-intercept	x	=

We are given the quadratic function:

\[
f(x) = x^2 - 20x - 800
\]

Let's go step by step to find the **vertex**, **y-intercept**, and **x-intercepts** of this parabola, and sketch the graph.

### 1. Vertex of the Parabola

The vertex form of a quadratic function is given by:

\[
f(x) = a(x - h)^2 + k
\]

However, we can find the vertex directly using the formula for the x-coordinate of the vertex:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

For the given quadratic function \( f(x) = x^2 - 20x - 800 \), the coefficients are:
- \( a = 1 \)
- \( b = -20 \)
- \( c = -800 \)

So, the x-coordinate of the vertex is:

\[
x_{\text{vertex}} = \frac{-(-20)}{2(1)} = \frac{20}{2} = 10
\]

To find the y-coordinate of the vertex, substitute \( x = 10 \) into the equation:

\[
y_{\text{vertex}} = f(10) = (10)^2 - 20(10) - 800 = 100 - 200 - 800 = -900
\]

So, the **vertex** is at \( (10, -900) \).

### 2. Y-Intercept

The y-intercept occurs where \( x = 0 \). To find the y-intercept, substitute \( x = 0 \) into the quadratic function:

\[
y_{\text{intercept}} = f(0) = 0^2 - 20(0) - 800 = -800
\]

So, the **y-intercept** is \( y = -800 \).

### 3. X-Intercepts

The x-intercepts occur where \( f(x) = 0 \). To find the x-intercepts, solve the equation:

\[
x^2 - 20x - 800 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values of \( a = 1 \), \( b = -20 \), and \( c = -800 \):

\[
x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(1)(-800)}}{2(1)}
\]

\[
x = \frac{20 \pm \sqrt{400 + 3200}}{2}
\]

\[
x = \frac{20 \pm \sqrt{3600}}{2}
\]

\[
x = \frac{20 \pm 60}{2}
\]

Now, solve for the two values of \( x \):

\[
x = \frac{20 + 60}{2} = \frac{80}{2} = 40
\]

\[
x = \frac{20 - 60}{2} = \frac{-40}{2} = -20
\]

So, the **x-intercepts** are \( x = -20 \) and \( x = 40 \).

### Summary of Key Points:

- **Vertex:** \( (10, -900) \)
- **Y-Intercept:** \( y = -800 \)
- **X-Intercepts:** \( x = -20, 40 \)

### Sketching the Graph:

- The parabola opens upwards since the coefficient of \( x^2 \) is positive (\( a = 1 \)).
- The vertex is at \( (10, -900) \), the lowest point on the graph.
- The y-intercept is \( (0, -800) \).
- The x-intercepts are \( (-20, 0) \) and \( (40, 0) \).

Would you like more details on graphing, or any specific part explained further?

### Related Questions:
1. How do you find the axis of symmetry for a quadratic function?
2. Can the vertex formula be used for any quadratic equation?
3. What is the significance of the discriminant in the quadratic formula?
4. How do transformations affect the graph of a quadratic function?
5. What is the importance of the sign of the \( a \)-term in a quadratic equation?

**Tip:** The x-coordinate of the vertex always lies on the axis of symmetry, which divides the parabola into two mirror-image halves.

Sketch the graph of the quadratic function. f(x) = x^2 − 20x − 800. Indicate the coordinates of the vertex, the y-intercept and the x-intercepts (if any). (If an answer does not exist, enter DNE. If there are multiple x-intercepts, enter your answers as a comma-separated list.)

Explore how to sketch the graph of the quadratic function f(x) = x^2 − 20x − 800. Learn to calculate the vertex, y-intercept, and x-intercepts using algebraic formulas. This guide is perfect for high school students learning quadratic functions.

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