Question

math 108 3.3 worksheet

1. complete the square to write the function in vertex form, y = a(x - h)^2 + k, and identify the vertex:

a) y - 1 = x^2 - 10x
b) y = 4x^2 + 24x + 13

Explanation:

Step1: Isolate the $x$ - terms for part a

Given $y - 1=x^{2}-10x$.

Step2: Complete the square for part a

Add and subtract $(\frac{-10}{2})^2 = 25$ on the right - hand side: $y-1=x^{2}-10x + 25-25$.
Rewrite as $y-1=(x - 5)^2-25$.
Then $y=(x - 5)^2+24$. The vertex is $(5,24)$.

Step3: Factor out the coefficient of $x^{2}$ for part b

Given $y = 4x^{2}+24x + 13$, factor out 4 from the first two terms: $y=4(x^{2}+6x)+13$.

Step4: Complete the square inside the parentheses for part b

Add and subtract $(\frac{6}{2})^2 = 9$ inside the parentheses: $y=4(x^{2}+6x + 9-9)+13$.
Rewrite as $y=4((x + 3)^2-9)+13$.
Expand: $y=4(x + 3)^2-36 + 13$.
So $y = 4(x + 3)^2-17$. The vertex is $(-3,-17)$.

Answer:

a) $y=(x - 5)^2+24$, vertex: $(5,24)$
b) $y = 4(x + 3)^2-17$, vertex: $(-3,-17)$