Answer:
In vertex form:
In standard form:
Step-by-step explanation:
The equation of a parabola in vertex form is
y = a (x - h)² + k,
where
a is the same as the a coefficient in the standard form,
h is the x-coordinate of the parabola vertex,
and k is the y-coordinate of the parabola vertex1. If you have the equation of a parabola in vertex form, then the vertex is at (h, k) and the focus is at (h, k + a/4)
Given vertex (h, k) = (-2, ) we get h = - 2 and k= 5
So on first pass we get the equation of the parabola as
[tex]y = a(x - -2)^2 + 5[/tex]
[tex]y = a(x + 2)^2 +5[/tex]
Given focus (h, k + a/4) = (-2, 6) we get
k + a/4 = 6
Substituting k = 5 we get
5 + a/4 = 6
a/4 = 6 - 5
a/4 = 1
a = 4
So the equation of the parabola in vertex form is
y = 4(x+2)² + 5
We can convert this to standard form as follows:
y = 4(x + 2)² + 5
y = 4(x² + 4x + 4) + 5 using the identity (x + a)² = x² + 2ax + a²
y = 4x² + 16x + 16 + 5
y = 4x² + 16x + 21
Answer:
[tex](x+2)^2=4(y-5)[/tex]
Step-by-step explanation:
Since the x-coordinate of the given vertex and focus of the parabola are the same, the parabola has a vertical axis of symmetry, indicating that it opens either upward or downward.
The standard form of a vertical parabola is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Standard form of a vertical parabola}}\\\\(x-h)^2=4p(y-k)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{Vertex:\;$(h,k)$}\\\phantom{ww}\bullet\;\textsf{Focus:\;$(h,k + p)$}\\\phantom{ww}\bullet\;\textsf{Directrix:\;$y=(k-p)$}\\\phantom{ww}\bullet\;\textsf{Axis of symmetry:\;$x=h$}\\\phantom{ww}\bullet\;\textsf{$p =$ distance between vertex and focus.}\end{array}}[/tex]
Given that the vertex is (-2, 5), then:
To find the value of p, we can use the distance between the vertex (-2, 5) and the focus (-2, 6), which is 1 unit in the y-direction, so:
Plug in h = -2, k = 5 and p = 1 into the formula:
[tex](x-(-2))^2=4(1)(y-5)[/tex]
Simplify:
[tex](x+2)^2=4(y-5)[/tex]
Therefore, the equation of the parabola in standard form is:
[tex]\Large\boxed{\boxed{(x+2)^2=4(y-5)}}[/tex]
