Respuesta :
Answer:
Vertex: (4, -2)
Vertex form: [tex]y = 2(x-4)^2 - 2[/tex]
Inverse function: [tex]y = (16 \pm\sqrt{16+8x})/4[/tex]
Step-by-step explanation:
The x-coordinate of the vertex can be calculated using the formula:
[tex]x_{vertex} = -b/2a[/tex]
Where a and b are coefficients of the quadratic equation in the form:
[tex]y = ax^2 + bx + c[/tex]
So, using a = 2 and b = -16, we have:
[tex]x_{vertex} = 16/4 = 4[/tex]
To find the y-coordinate of the vertex, we calculate y using the x-coordinate of the vertex:
[tex]y=2*4^2-16*4+30[/tex]
[tex]y = -2[/tex]
So the vertex is (4, -2)
The vertex form is given by:
[tex]y = a(x-h)^2 + k[/tex]
Where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex. Therefore, we have that:
[tex]y = 2(x-4)^2 - 2[/tex]
To create an inverse function, we switch x by y and vice-versa:
[tex]x=2y^2-16y+30[/tex]
[tex]2y^2-16y+30 - x = 0[/tex]
Then, using Bhaskara's formula, we have:
[tex]\Delta = b^2 -4ac = (-16)^2 - 4*2*(30-x) = 16 + 8x[/tex]
[tex]y = (-b \pm\sqrt{\Delta})/2a[/tex]
[tex]y = (16 \pm\sqrt{16+8x})/4[/tex]
It's important to say that this inverse function is not really a function, because one value of x gives two values of y.
