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Question 4
How are the coordinates of the extreme value and the equation y=(x-1)^2-8 in the form y = (x – h)2 – c related?
How are the coordinates of the extreme value and the equation y=(x-1)^2-8 in the form y = (x – h)2 – c related?
Answers
Answered by
GPT-4o mini
The equation given is \( y = (x - 1)^2 - 8 \). This is a quadratic function in vertex form, which is generally expressed as:
\[
y = (x - h)^2 - k
\]
where \( (h, k) \) represents the coordinates of the vertex (or extreme value) of the parabola.
In your equation:
- \( h = 1 \)
- \( k = -8 \)
Thus, the vertex of the parabola represented by the equation \( y = (x - 1)^2 - 8 \) is located at the point \( (1, -8) \).
The vertex is the point where the parabola reaches its minimum (since the coefficient of \( (x - 1)^2 \) is positive, indicating that it opens upwards). Therefore, the extreme value (minimum) of the function is \( -8 \) at \( x = 1 \).
In summary, the coordinates of the extreme value (the vertex) of the parabola described by your equation are related to the parameters \( h \) and \( k \) in the vertex form \( y = (x - h)^2 - k \). Specifically, \( h \) gives the x-coordinate of the extreme value, and \( -k \) gives the y-coordinate of the extreme value. Thus, in this case:
- The coordinates of the extreme value are \( (h, k) = (1, -8) \).
\[
y = (x - h)^2 - k
\]
where \( (h, k) \) represents the coordinates of the vertex (or extreme value) of the parabola.
In your equation:
- \( h = 1 \)
- \( k = -8 \)
Thus, the vertex of the parabola represented by the equation \( y = (x - 1)^2 - 8 \) is located at the point \( (1, -8) \).
The vertex is the point where the parabola reaches its minimum (since the coefficient of \( (x - 1)^2 \) is positive, indicating that it opens upwards). Therefore, the extreme value (minimum) of the function is \( -8 \) at \( x = 1 \).
In summary, the coordinates of the extreme value (the vertex) of the parabola described by your equation are related to the parameters \( h \) and \( k \) in the vertex form \( y = (x - h)^2 - k \). Specifically, \( h \) gives the x-coordinate of the extreme value, and \( -k \) gives the y-coordinate of the extreme value. Thus, in this case:
- The coordinates of the extreme value are \( (h, k) = (1, -8) \).