Algebra 2 H.

This is only a small portion of what we learned this year!!

Decibel Level (equation)

Decibels in dB

Neutral Substance

pH = 7

Acidic Substance

pH < 7

Alkaline Substance

pH > 7

pH scale

0 to 14

[𝐻⁺]

the concentration of hydrogen ions (in moles/liter or mol/L) of the substance.

Equation for the pH of a Substance

The formula for interest compounded n times a year

The formula for interest compounded continuously

Exponential Growth Formula

Exponential Decay Formula

Square Root Property

The square root property states that if x^2 = k (where k is a real number), then x = ±√k.

This method is used to solve quadratic equations by taking the square root of both sides, ensuring both the positive and negative roots are included

a³+b³

(a+b)(a²-ab+b²)

a³-b³

(a-b)(a²+ab+b²)

multiplicity

the number of times a specific value (or factor) appears as a root/zero in a given polynomial or multiset.

Discriminant

D = b² - 4ac

D < 0

No real roots; 2 imaginary roots

D = 0

One real root

D > 0

2 real roots

For f(x) = xⁿ where n is and even number.
What happens when you change n?

As n increases, the graph becomes flatter and closer to the x-axis.

As n decreases, the graph becomes steeper, shooting up more rapidly toward infinity.

End behavior of an odd function with a positive leading coefficient

f(x) → +♾ as x → +♾

f(x) → -♾ as x → -♾

End behavior of an odd function with a negative leading coefficient

f(x) → -♾ as x → +♾

f(x) → +♾ as x → -♾

End behavior of an even function with a positive leading coefficient

f(x) → +♾ as x → +♾

f(x) → +♾ as x → -♾

End behavior of an even function with a negative leading coefficient

f(x) → -♾ as x → +♾

f(x) → -♾ as x → -♾

Direct Variation

y varies directly as x

y = kx

k is the constant of variation or the constant of proportionality

Inverse Variation

y varies inversely as x

y = k/x

k is the constant of variation or the constant of proportionality

Joint Variation

y varies jointly as the other variables

y = kxz

y = kxzw

ect.

k is the constant of variation or the constant of proportionality

Finding the nth root of (aⁿ)

If n is an even positive integer, then the nth root of (aⁿ) is |a|

If n is an odd positive integer, then the nth root of (aⁿ) is a

Quotient Rule for Radicals

Product Rule for Radicals

Solving a Radical Equation

1. Isolate one radical on one side of the equation

2. Raise each side of the equation to a power equal to the index of the radical and simplify

3. If the equation still contains a radical term, repeat 1 and 2. If not ssolve the equation

4. Check all proposed solution in the original equation

Imaginary Unit

The imaginary unit, written i, is the number whose square is -1.

i² = -1 and i = √(-1)

Complex Numbers

A no. that can be written in the form a + bi where a and b are real nos.

Sum of Complex Numbers

(a + bi) + (c + di) = (a + c) + (b + d)i

Difference of Complex Numbers

(a + bi) - (c + di) = a + bi - c - di = (a - c) + (b - d)i

Complex Conjugates

Ex: (a + bi) and (a - bi)

(a + bi)•(a - bi) = a² + b²

Square Root Property

If b is a real number and if a² = b, then a = ±√b

Quadratic Formula

Used when quadratic is in standard form: ax² + bx + c = 0

Solving a Quadratic Equation

1. If eqn. is in the form (ax + b)² = c, use the square root property and solve. If not, go to step 2

2. Write the eqn. in standard form: ax² + bx + c = 0

3. Try to factor. If it can’t be factored, go to step 4

4. Solve using the quadratic formula

Rational Zero Thm.

possible rational zeros = (factors of the constant term) / ( factors of the leading coefficient)

The Fundamental Thm. of Algebra

If f(x) is a polynomial of degree n, where n ≥ 1, then the eqn. f(x) = 0 has a least one complex root

One-to-One Function

each input (x-value) corresponds to only one output (y-value), and each output (y-value) corresponds to only one input (x-value)

Needs to pass both the horizontal line test and the vertical line test

A function needs to be a one-to-one to have and inverse function

Exponential function

f(x) = b^x

b > 0, b ≠ 1, and x is a real no.

• one-to-one function

• y-int: (0,1)

• no x-int

• d: all real nos

• r: y > 0

Uniqueness Property of b^x

Let b > 0 and b ≠ 1. Then b^x = b^y is equivalent to x = y.

Logarithmic Definition

If b > 0 and b ≠ 1, then…

y = log_bx means x = b^y

for every x > 0 and every real number y.

Properties of Logarithms

If b is a real number, b > 0, and b ≠ 1, then…

1. log_b1 = 0

2. log_bb^x = x

3. b^log_bx = x

Logarithmic Function

If x is a positive real number, b is a constant positive real number, and b ≠ 1, then a logarithmic function can be defined by:

f(x) = log_bx

The domain of f is the set of positive real numbers, and the range of f is the set of all real numbers

• one-to-one function

• x-int: (1,0)

• no y-int

• d: x > 0

• r: all real nos

Product Property of Logarithms

If x, y, and b are positive real numbers and b ≠ 1, then…

log_bxy = log_bx + log_by

Power Property of Logarithms

If x and b are positive real numbers, b ≠ 1, and r is a real number, then…

log_bx^r = r • log_bx

Quotient Property of Logarithms

If x, y, and b are positive real numbers and b ≠ 1, then…

log_b(x/y) = log_bx - log_by

Change of Base Property

If a, b, and c are positive real numbers and neither b nor c is 1, then…

log_ba = (log_ca) / (log_cb)

Logarithm Property of Equality

Let a, b, and c be real numbers such that log_ba and log_bc are real numbers and b ≠ 1, then…

log_ba = log_bc is equivalent to a = c

Rational Functions

f(x) = p(x) / q(x)

q(x) ≠ 0

Horizontal Asymptotes of Rational Functions

Graphing Rational Functions

f(x) = p(x) / q(x)

Where p and q are polynomial functions with no common factors

1. Find the y-int. (if there is one)

2. Find the x-ints. (if there are any) by solving p(x) = 0

3. Find any VA(s) by solving q(x) = 0

4. Find the HA (if there is one) by using the rule for the HAs of rational functions

5. Plot at least 1 point between and beyond each x-int and VA

6. Use the info obtained prev. to graph the function

Horizontal Parabolas

Standard form: x = a(x - h)² + k

Vertex: (h, k)

If a>0, then opens to the right

If a<0, then opens to the left

Axis of Symmetry: y = k

Circle

(x - h)² + (y - k)² = r²

Center: (h, k)

Radius: r