Question

Write each function in vertex form and give the vertex. If ( a neq 1 ), factor it outside so it becomes ( a(x^2 + bx) ). Complete the square by taking half of ( b ) and squaring it. This is the new "c". Add this to ( x^2 + bx ). Subtract ( a times c ) from the end of the equation. Factor the trinomial and simplify the end of the equation.

a) ( f(x) = a(x - h)^2 + k )
b) ( f(x) = a(x + h)^2 - k )
c) ( f(x) = a(x - h)^2 - k )
d) ( f(x) = a(x + h)^2 + k )

Answer 1

To write a quadratic function in vertex form, follow the steps: factor a outside, find the new c value, add it to the equation, subtract a × c, factor the trinomial, and simplify. Options (a), (b), (c), and (d) correspond to different vertex forms.

Step-by-step explanation:

To write a quadratic function in vertex form, you have to complete the square by following these steps:

1. If a is not equal to 1, factor it outside the parentheses as a(x^2 + bx).
2. Take half of b and square it to find the new value of c.
3. Add this new c to x^2 + bx.
4. Subtract a × c from the end of the equation.
5. Factor the resulting trinomial and simplify the equation.
6. Write the equation in the form of a(x - h)^2 + k or a(x + h)^2 - k, depending on the signs of the terms.

For option (a), the vertex form is f(x) = a(x - h)^2 + k. For option (b), the vertex form is f(x) = a(x + h)^2 - k. For option (c), the vertex form is f(x) = a(x - h)^2 - k. For option (d), the vertex form is f(x) = a(x + h)^2 + k.