Question

Write an equation for a quadratic function in vertex form with

Vertex (-1, 6) that passes through the point (-3,4).

Answer 1

`\displaystyle f(x)=-(1)/(2)(x+1)^2+6`

Explanation:

We want to write the equation for a quadratic function in vertex form with vertex at (-1, 6) that passes through the point (-3, 4).

Vertex form is given by:

`\displaystyle f(x)=a(x-h)^2+k`

Where (h, k) is the vertex and a is the leading coefficient.

Since our vertex is at (-1, 6), h = -1 and k = 6. Substitute:

`f(x)=a(x-(-1))^2+(6)`

Simplify:

`f(x)=a(x+1)^2+6`

Next, since the quadratic passes through the point (-3, 4), f(x) = 4 when x = -3. Substitute:

`(4)=a(-3+1)^2+6`

Solve for a. Simplify:

`-2=a(-2)^2`

Hence:

`\displaystyle -2=4a\Rightarrow a=-(1)/(2)`

Therefore, our function in vertex form is:

`\displaystyle f(x)=-(1)/(2)(x+1)^2+6`