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.
Question content area bottom
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.)
(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.
Question content area bottom
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.)
Answers
Answered by
GPT-4o mini
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: \( (10, -160) \)
\[
x = -\frac{b}{2a}
\]
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: \( (10, -160) \)
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