Question 619129: I have been stuck on this problem for days can someone please help me use completing the square to describe the graph f(x)=40-12x-x^2 support your answer graphically.
Answer by mathgranny(13)   (Show Source):
You can put this solution on YOUR website! i have been stuck on this problem for days can someone please help me.use
completing the square to describe the graph f(x)=40-12x-x^2.support your answer graphically.
First review what a perfect square is. It is a trinomial which can be obtained by squaring a binomial( multiplying a binomial by itself.
For example a binomial has 2 terms. 2x^2 + 2x
or 4x + 5
When you have a trinomial with an x^2 term and
one x term
and one constant(number)
then the kind of binomial we
are looking to square is of the
form bx + c.
Let's square some binomials.
a) (x + 2) ^2 is (x + 2)(x + 2) = x ^2 + 4x + 4
b) (x + 3 ) ^2 = (x + 3)(x + 3) = x ^2 + 6x + 9
c) (x + 4) ^2 = (x + 4)(x +4) = x^2 + 8x +16
d) AND (x + 5) ^ 2 = (x+5)(x+5) = x^2 +10x + 25
I'm sensing a pattern.
a) 2(2) =4, 2^ 2 =4
b) 3(2) = 6, 3^2 =9
c) 4(2) =8 , 4^2 =8
d) 5(2 = 10, 5^2 =25
Now practice squaring binomials without writing each one twice as aabove.
(x + 6)^2 = X^2 + ___x + ___
For the x coefficient, double the 6.
For the constant term, square the 6
so (x + 6) ^ 2 = x^2 + 12x + 36
x^2 +12x + ? to make a perfect square?
Take 1/2 of 12 and square it . 6^2 = 36.
Make this binomial into a perfect square.
x^2+ 16x + ? ( 1/2 of 16 = 8. 8^2 =64)
x^2 +16x + 64
We cannot have a - in front of the x, because squaring a number
never produces a negative.
Write 40 -12x - x^2 in standard form as
A. -x^2 -12x + 40 = y(or f(x))
Next we replace y with 0.
Why? Because points with y coordinate = 0
are on the x-axis., giving us the
x-intercepts.
Once we know the position of the intercepts we
find the line of symmetry that is
halfway between these points.
A. -x^2 -12x +40 = 0
-x^2 -12x = -40 (Multiply by -1.
B. x^2 + 12x = +40
Now what do we add to left
side to complete the square?
We add 1/2(12) and square it, getting 36. Add to BOTH sides.
B. x^2 +12x +36 = 40 +36
(x + 6)^2 = 76. taking sq. roots:
x + 6 = sq. root of 76
x + 6 = 8.7 0r x + 6 = -8.7
x = 2.7 and x = -14.7
Intercepts are (2.7, 0) and ( -14.7, 0)
Line of symmetry is x = -6
The vertex is on this line .
The x-coordinate is -6, and y coordinate is f(-6)
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