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: \( (10, -160) \)