Question

Draw the graph of the function.
y equals negative x squared minus 80 x plus 2000
Question content area bottom left
Part 1
Use the graphing tool to graph the equation.

Answer 1

To graph the quadratic function \( y = -x^2 - 80x + 2000 \), follow these steps:

### Step 1: Identify the Components of the Function
This is a quadratic function in standard form \( y = ax^2 + bx + c \), where:
- \( a = -1 \) (the parabola opens downwards),
- \( b = -80 \),
- \( c = 2000 \).

### Step 2: Find the Vertex
The vertex of the parabola can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{-80}{2 \cdot -1} = \frac{80}{-2} = -40
\]

Now substitute \( x = -40 \) back into the equation to find the \( y \) coordinate of the vertex:
\[
y = -(-40)^2 - 80(-40) + 2000
\]
\[
= -1600 + 3200 + 2000 = 3600
\]

So, the vertex is at \( (-40, 3600) \).

### Step 3: Determine the Axis of Symmetry
The axis of symmetry is the line \( x = -40 \).

### Step 4: Find the y-intercept
The y-intercept occurs when \( x = 0 \):
\[
y = -0^2 - 80(0) + 2000 = 2000
\]
So, the y-intercept is at \( (0, 2000) \).

### Step 5: Find x-intercepts
Set \( y = 0 \) to find the x-intercepts:
\[
0 = -x^2 - 80x + 2000
\]
Rearranging gives:
\[
x^2 + 80x - 2000 = 0
\]
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{-80 \pm \sqrt{80^2 - 4(1)(-2000)}}{2(1)}
\]
\[
= \frac{-80 \pm \sqrt{6400 + 8000}}{2}
\]
\[
= \frac{-80 \pm \sqrt{14400}}{2}
\]
\[
= \frac{-80 \pm 120}{2}
\]
Calculating the two potential solutions:
\[
x = \frac{40}{2} = 20 \quad \text{and} \quad x = \frac{-200}{2} = -100
\]
Thus, the x-intercepts are at \( (20, 0) \) and \( (-100, 0) \).

### Step 6: Plot the Points and Sketch the Graph
1. **Vertex**: \( (-40, 3600) \)
2. **Y-Intercept**: \( (0, 2000) \)
3. **X-Intercepts**: \( (20, 0) \) and \( (-100, 0) \)

You would plot these points on graph paper or a graphing tool and then sketch the parabola opening downwards, ensuring it passes through these significant points.

### Final Graph
- The vertex is the highest point at \( (-40, 3600) \).
- The y-intercept is below the vertex at \( (0, 2000) \).
- The x-intercepts are to the right and left \( (20, 0) \) and \( (-100, 0) \).

If you have access to graphing software or tools, you can enter the equation directly to visualize it more accurately.