Question

which values for h and k are used to write the function $f(x) = x^2 + 12x + 6$ in vertex form?

• $h=6, k=36$
• $h=-6, k=-36$
• $h=6, k=30$
• $h=-6, k=-30$

Explanation:

Step1: Recall vertex form

Vertex form: $f(x)=(x-h)^2+k$

Step2: Complete the square

Take coefficient of $x$: $12$. Half it: $\frac{12}{2}=6$. Square it: $6^2=36$.
Rewrite function:
$$f(x) = (x^2+12x+36) + 6 - 36$$

Step3: Simplify the expression

$$f(x)=(x+6)^2 - 30$$
Rewrite to match vertex form: $f(x)=(x-(-6))^2 + (-30)$

Step4: Identify h and k

Compare to $f(x)=(x-h)^2+k$: $h=-6$, $k=-30$

Answer:

h=-6, k=-30