Exercise 51 Page 208

We are given a quadratic function and its vertex, and want to find the values of a and b. Function & Vertex y=ax^2+bx+8 & (2,-4) To do so, we will use the vertex form of a quadratic function. y=a(x- h)^2+ kIn this form, the vertex is ( h, k). Since we know the vertex of our function is ( 2, -4), we can substitute 2 for h and -4 for k. y=a(x- 2)^2+( -4) ⇕ y=a(x-2)^2-4 Let's simplify this equation.

y=a(x-2)^2-4

ExpandNegPerfectSquare

(a-b)^2=a^2-2ab+b^2

y=a ( x^2-2(x)(2)+2^2 ) -4

Simplify power and product

y=a ( x^2-4x+4 ) -4

Distr

Distribute a

y=ax^2-4ax+4a-4

Now, we can compare the given function with the one we obtained, term by term.

	y=ax^2+bx+8	y=ax^2+(-4a)x+(4a-4)
Quadratic Coefficient	a	a
Linear Coefficient	b	-4a
Independent Term	8	4a-4

We can find the value of a by equating the constant terms.

8= 4a-4

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Simplify

AddEqn

LHS+4=RHS+4

12=4a

DivEqn

.LHS /4.=.RHS /4.

3 = a

RearrangeEqn

Rearrange equation

a=3

Finally, from the linear coefficient, we know that b= -4a. We will find the value of b by substituting 3 for a. b=-4(3) ⇔ b=-12