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How to “complete the Square”for f(x) = 3x2 + 27x - 24:

1. Move anything w/o an x to the leftf(x) + 24 = 3x2 + 27x

2. Make the number in front of x2 equal to 1(f(x) + 24)/3 = x2 + 9x

3.Take the number in front of x, divide by 2 and then square the result. (9/2)2

4. Add the result of step 3 to both sides. (f(x) + 24)/3 + (9/2) 2 = x2 + 9x + (9/2)2

5.Factor the right side. (f(x) + 24)/3 + 20.25 = [x - (9/2)]2

6. Move everything back to the right and now you have the standard quadratic function equation. f(x) = 3[x - (9/2)]2 – 84.75

What is the Standard form of aQuadratic Function?

f(x) = a(x-h)2+k am0

The graph of f is a parabola

with vertex at point (h,k)

if a>0, it opens UPward;

if a<0, it opens DOWNward

In a Standard Quadratic Function, What are the formulas for the points on the Vertex (h, k) ?

(h, k)

=(-b , c-b2) 2a 4a

What is the quadratic Formula?

X= -b kpb2-4ac

2a

For standard quadratic equations:

ax2 + bx + c = 0 am0

4 Steps For Finding Inverse Function:

- 1. Change the function notation : f(x) y
- 2. Change all the xs to ys and ys to xs
- 3. Solve for Y
- 4. Replace y with f-1(x)

Domain Questions

- Does the function have a denominator?
- Does the function have a square or even root?
- Does the function have a log or ln in it?
- Did the function arise from finding an inverse?
- Is this a “real world” problem?

Shifts(addition or subtraction to the function)

- Horizontal shifts (inside parentheses)
- f(x + c) means the vertex moves left ( ) c units
- f(x - c)
- means the vertex moves right ( ) c units

- Vertical Shifts (outside parentheses)
- f(x)+ cmeans the vertex moves up c
- f(x) - c
- means the vertex moves down c

Stretch/Shrink(multiplication or division of the function)

- Vertical(outside parentheses)

c f(x)

- When c > 1 then stretch
- When 0< c < 1 then shrink

- Horizontal(inside parentheses)

f(cx)

- When c > 1 then shrink
- When 0< c < 1 then stretch

We use COMPLEX NUMBERS to remove negative numbers from inside radicals.

p-1 = i and i2 = -1

How do you figure the exponents of i ?

Divide the exponent by 4

If it divides evenly by 4, then the answer is 1.

The repeating pattern is:

-1, -i, 1, i

i2 =-1i3=-ii4=1i5=i

What is a complex conjugate?

Fora+bi, it isa-bi

How do you divide by a complex number?

Multiply both numerator and denominator

by the complex conjugateof the denominator.

3-2i(3-2i)(4+5i)

4-5i = (4-5i)(4+5i)

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