Question
What is the vertex form of the graphed equation? (-4,-2)
y = - (x -
)2 +
y = - (x -
)2 +
Answers
GPT-4o mini
To write the equation of a parabola in vertex form, we use the formula:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola and \( a \) determines the direction and width of the parabola.
Given the vertex \( (-4, -2) \), we can substitute \( h \) and \( k \) into the equation:
- \( h = -4 \)
- \( k = -2 \)
So, we can substitute these values into the vertex form:
\[
y = a(x - (-4))^2 - 2
\]
or simplified:
\[
y = a(x + 4)^2 - 2
\]
You mentioned that the equation starts with \( y = - (x - \ldots)^2 + \ldots \). It looks like you want the parabola to open downwards since the leading coefficient appears to be negative (-).
Thus, the equation now becomes:
\[
y = -a(x + 4)^2 - 2
\]
To fully specify the equation, we need to know the value of \( a \). The value of \( a \) can be determined based on another point on the parabola or more information about its width. If you have that information, please provide it; otherwise, we can leave \( a \) as a variable.
So the vertex form of the equation will be:
\[
y = -a(x + 4)^2 - 2
\]
where \( a > 0 \) makes the parabola wider, while \( a < 0 \) makes it narrower (if you later specify a point on the graph or the value of \( a \), we can determine the exact form).
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola and \( a \) determines the direction and width of the parabola.
Given the vertex \( (-4, -2) \), we can substitute \( h \) and \( k \) into the equation:
- \( h = -4 \)
- \( k = -2 \)
So, we can substitute these values into the vertex form:
\[
y = a(x - (-4))^2 - 2
\]
or simplified:
\[
y = a(x + 4)^2 - 2
\]
You mentioned that the equation starts with \( y = - (x - \ldots)^2 + \ldots \). It looks like you want the parabola to open downwards since the leading coefficient appears to be negative (-).
Thus, the equation now becomes:
\[
y = -a(x + 4)^2 - 2
\]
To fully specify the equation, we need to know the value of \( a \). The value of \( a \) can be determined based on another point on the parabola or more information about its width. If you have that information, please provide it; otherwise, we can leave \( a \) as a variable.
So the vertex form of the equation will be:
\[
y = -a(x + 4)^2 - 2
\]
where \( a > 0 \) makes the parabola wider, while \( a < 0 \) makes it narrower (if you later specify a point on the graph or the value of \( a \), we can determine the exact form).