Find the equation for the parabola with the given vertex that passes through the given points.
vertex (-4,2)
point (-3,0)
please show work!!!
You should be familiar with the vertex form of a parabola
y =a(x-p)^2 + q , with vertex (p,q)
so for yours, vertex is (-4,2) , so
y = a(x+4)^2 + 2
sub in (-3.0)
0 = a(1)^2 + 2
a = -2
y = -2(x+4)^2 + 2
To find the equation of a parabola given the vertex and a point, we can use the standard form equation of a parabola:
y = a(x - h)^2 + k,
where (h, k) is the vertex and 'a' is a coefficient determining the shape of the parabola.
Given the vertex (-4, 2), we can substitute these values into the equation:
y = a(x - (-4))^2 + 2
Simplifying further:
y = a(x + 4)^2 + 2
Now we need to determine the value of 'a'. To do this, we'll use the other given point (-3, 0).
Substituting this point into the equation, we get:
0 = a(-3 + 4)^2 + 2
Simplifying:
0 = a(1)^2 + 2
0 = a + 2
Subtracting 2 from both sides:
-2 = a
Now we know the value of 'a'. Substituting it back into the equation, we get:
y = -2(x + 4)^2 + 2
Therefore, the equation of the parabola with the given vertex (-4, 2) and passing through the point (-3, 0) is:
y = -2(x + 4)^2 + 2.