What is the vertex form of y=(-x-1)(x+7)?

1 Answer
Dec 23, 2016

"Vertex form "->" "y=-1(x color(magenta)(-3))^2color(blue)(+2)

"Vertex"->(x,y)=(3,2)

Explanation:

First return this to the form of y=ax^2+bx+c

y=color(blue)((-x-1))color(brown)((x+7))

Multiply everything in the right hand bracket by everything in the left.

y=color(brown)(color(blue)(-x)(x+7)color(blue)(" "-1)(x+7))

y=-x^2+7x" "-x-7

y=-x^2+6x-7.............................Equation(1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Write as: y=-1(x^2-6x)-7+k
The k corrects the error this process introduces.

Move the power from x^2 to the outside of the btackets

y=-1(x-6x)^2-7+k

Halve the 6 from 6x

y=-1(x-3x)^2-7+k

Remove the x from the 3x

y=-1(x-3)^2-7+k......................Equation(1_a)
........................................................................................
Dealing with the error
If you were to expand the brackets and multiply by the -1 you have the value of (-1)(-3)^2 =-9. Looking back at Equation(1) you will observe that this value is not in it. So we have to remove the -9

Set -9+k=0 => k=9
.....................................................................................
Substitute for k" in "Equation(1_a)

y=-1(x-3)^2-7+k color(green)(" "->" "y=-1(x-3)^2-7+9)

y=-1(x color(magenta)(-3))^2color(blue)(+2)

x_("vertex")=(-1)xx color(magenta)((-3)) =+3
y_("vertex")=color(blue)(+2)

"Vertex"->(x,y)=(3,2)

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