Question

Consider the function y equals 2 x squared minus 40 x plus 40.
​(a) Find the​ x-coordinate of the vertex of the graph.
​(b) Set the viewing window so that the​ x-coordinate of the vertex is near the center of the window and the vertex is​ visible, and then graph the given equation.
​(c) State the coordinates of the vertex.
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Part 1
​(a) The​ x-coordinate of the vertex of the graph is xequals
  
enter your response here.
​(Type an integer or a simplified​ fraction.)

Answer 1

To find the x-coordinate of the vertex of the quadratic function \( y = 2x^2 - 40x + 40 \), we can use the vertex formula:

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

In our function, \( a = 2 \) and \( b = -40 \). Plugging these values into the formula gives:

\[
x = -\frac{-40}{2 \times 2} = \frac{40}{4} = 10
\]

### Part 1 (a)
The x-coordinate of the vertex of the graph is \( x = 10 \).

### Part 2 (b)
To set the viewing window so that the x-coordinate of the vertex is near the center and the vertex is visible, we can choose a window that centers around \( x = 10 \). A reasonable choice could be:

- **x-axis range**: From \( x = 0 \) to \( x = 20 \) (giving a range of 20)
- **y-axis range**: We need to compute the y-coordinate of the vertex to determine the appropriate range.

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

\[
y = 2(10)^2 - 40(10) + 40
\]
\[
y = 2(100) - 400 + 40
\]
\[
y = 200 - 400 + 40 = -160
\]

Because the vertex falls at \( (10, -160) \), we can set a reasonable y-axis range that includes this value. Let's choose:

- **y-axis range**: From \( y = -200 \) to \( y = 20 \).

This should ensure that the vertex is visible on the graph.

### Part 3 (c)
The coordinates of the vertex are:
\[
(10, -160)
\]

In summary:
- (a) \( x = 10 \)
- (b) Set viewing window: \( x: [0, 20] \), \( y: [-200, 20] \)
- (c) Vertex coordinates: \( (10, -160) \)