Amy's Reflection #17

ok since we're having exams i think im gonna post something from the begininng of the year..ya know things we might have forgotten to do...

Completing the Square:

You can use completing the square to solve a quadratic equation when factoring doesn’t work. This method can only work when 1 is the coefficient of x².

For example:

x² + 6x - 2 = 0

x² + 6x = 2

x² + 6x + 9 = -2 + 9

(x + 3)² = 7

x + 3 = √7

x = -3 ± √7

(-3 + √7,0) (-3 -√7,0)

Rational Root therom:

Example: f(x)= 2x^3 + 3x^2 - 8 + 3

Step 1: find all possible roots..
p: factors of 3: 1, -1, 3, -3
q: factors of 2: 1, -1, 2, -2

*p is the leading constant term & q is the leading coefficient
possible roots are (p/q): 1, -1, 1/2, -1/2, 3, -3, 3/2, -3/2

Step 2: plug roots in calc & the zeros will be: 1, 1/2, -3

Step 3: synthetic division: (x - 1) (2x^2 + 5x + 3)

Step 4: slove further (factor): (x - 1) (2x^2 + 5x + 3)= (x - 1) (2x - 1) (x + 3)

Answer: x = 1, 1/2, -3

Domain & Range of functions:

Polynomials-domain of all polynomials is (−∞, ∞).

Fractions-you set the bottom to zero, solve for x, and then set up intervals

Square Roots-domain: set the inside = to zero, then set a # line, try values on either side of each #, and get ride of the negatives-range:graph

Absolute Value-domain: (- ∞ , + ∞)-range: [0 , + ∞)

How to Find the Inverse of a Function:

• Replace f(x) with y
• Reverse the roles of x and y
• Solve for y in terms of x
• Replace y with f-1(x)
• check: should equal to x

Example:

f(x) = √x + 4

(x)^2 = (√y + 4)^2

x^2 = y + 4

y = x^2 - 4

f-1(x) = (x^2 - 4)

f(f-1(x)) = f(x^2 - 4) = √(x^2 - 4) + 4 = x

f-1(f(x)) = f-1(√x + 4) = (√x + 4)^2 - 4 = x + 4 - 4 = x

Logarithm Properties:

• logb MN = logb M + logb N
• logb M/N = logb M - logb N
• logb M^K = K logb M
• logb b^k = k
• b^logb^k = k

Changing Bases: (Done when you can't solve a log)

• Rewrite it as an exponential
• Take the log of both sides
• Move the variable to the front
• then solve

Example:

log5 10 = x

5^x = 10

log 5^x = log 10

x log 5 = 1

x = 1/log 5

i hope this helped y'all remember some of this stuff...i still kinda need help with the graphing in chapter eight..like the question on the test we took monday...can someone help me with that??