UNIT 1Intro to Algebra II

NOTES

• Like Terms: terms in an algebraic expression or equation whose variable AND exponents are the same

• When we combine Like Terms by addition or subtraction, we DO NOT change the variable or its exponent, just the coefficient out front.

NOTES

• When we multiply terms in a polynomial expression:• Multiply the leading

coefficients.• Add the exponents of like

bases.

Factoring Perfect Square Quadratics

• Perfect square trinomial: a polynomial expression whose two factors are identical

• Difference of squares: two terms that are squared and separated by a subtraction sign.• The two factors are identical

except they have a different sign.

Factoring Perfect Square Quadratics

• Greatest Common Factor: in a set of algebraic terms, it is the highest number and lowest power shared by the terms

Linear Functions!!!• Linear functions are straight lines.

• Slope-intercept form of a linear function is

y = mx + b

WHEREm = slopeb = y-intercept(x, y) = points on the graph

Inverse Function• INVERSE FUNCTION: a function

obtained by expressing the dependent variable of one function as the independent variable of another.

• The BIG IDEA of inverse functions is that they undo each other.

• Graphically, the inverse function is a function reflected over the line y = x.

Inverse Function• To find the inverse of a

function:

1. Switch “x” and “y”.

2. Solve the equation for “y”.

3. Substitute f-1(x) in for “y”.

•Complex number: a number expressed in the form

where and are real numbers and i is imaginary.

•The number , so

Complex Numbers!

UNIT 2Quadratic Functions

Properties of Quadratics•An Extreme is a maximum or minimum of a

quadratic function. We also know this as the vertex.

•We can find the vertex of a quadratic function using the formula:

)•On the y-axis, the value of x is “0”. SO, we

can find the y-intercept of a quadratic function by plugging in “0” for x.

Properties of Quadratics•The Axis of

Symmetry is the line that cuts a quadratic function directly in half.

•The Axis of Symmetry is the vertical line that runs through the “x” coordinate of the vertex.

Properties of QuadraticsWe can find the x-intercepts of a quadratic function 3 ways:1. Using the quadratic formula:

2. By factoring3. By using the “2nd trace” button and “zero” button on our calculator

The Quadratic Formula NOTES

•The DISCRIMINANT of the quadratic formula tells us all about the types of solutions the function has.

•The DISCRIMINANT of the quadratic formula is:

When , there are 2 real solutions

When there is 1 real solution

When there are 2 NONREAL solutions

Quadratic Transformations• Transformations of Functions:

how functions move right/left, up/down, stretch/shrink, and reflect

• Parent Function: a function before it transforms• Today, the parent function was

always This was the original graph before it moved.

Quadratic Transformations• The Vertex Form of the quadratic

function is:

• When “k” is positive, the graph translates up. When “k” is negative, the graph translates down.

• When “h” is positive, the graph translates left. When “h” is negative, the graph translates right.

Quadratic Transformations• The Vertex Form of the quadratic

function is:

• When the graph vertically stretches.

• When the graph shrinks vertically.

• When “a” is negative, the graph reflects over the x-axis.

Quadratic Transformations• The Vertex Form of the quadratic

function is:

• NOTICE: (-h, k) is the vertex of the Quadratic Function!!!

UNIT 3Exponential & Logarithmic Functions

Exponential Functions

• An Exponential Function is a function with a base that is a number and an exponent that is a variable.

• While a linear function increases by a constant rate (slope), an exponential function increases by a constant percentage.

Exponential Functions

Exponentials FunctionsThe Standard Form for an Exponential Function:

WHERE:a = the initial value (starting value)b = the growth factor…this is the BASE(x, y) = points on the graph

Growth Factor (the BASE)• b = 1• When b is greater than 1 = growth• When b is less than 1 = decay

Graphing Exponentials!

• We can find the y-intercept by plugging in “0” for “x”

• We can find x-intercepts by plugging in “o” for “y” and solving for “x” OR using our calculator!

• End behavior describes how a function behaves as “x” values get really big and really small.

Continuous Growth/Decay

Compound Interest

Properties of Logarithms•Product Property:

•Quotient Property:

•Power Property:

UNIT 4Regression Analysis

Regression Analysis - Notes1. Make one variable “x” and one “y”.2. Hit “Stat”, then “Edit”. Put your “x” values in

L1. Put your “y” values in L2.3. Make sure you turn on Stat Plot and hit

“Zoom”, then “9” to see your data.4. Decide if the data is linear, quadratic, or

exponential.5. Find the line of best fit by hitting “Stat”, then

“Calc”, then the type of function (LinReg, QuadReg, or ExpReg)

6. Answer the question being asked using your knowledge of the function.

UNIT 5Polynomial Functions

Factoring Polynomials

UNIT 6Radicals, Absolute Value, & Circles

Rational Exponent Properties1. When we multiply like bases, ADD

exponents.2. When we divide like bases,

SUBTRACT exponents.3. When we raise an exponent to

another exponent, distribute the exponent.

4. You can change the sign of an exponent by flipping its place in the fraction.

Circles!!!• A circle is a set of points in a plane

equidistant from a fixed point, called the center.

• The General Form of a Circle:

• Center-Radius OR Standard Form of a Circle:

WHERE (h, k) is the center of the circle and “r” is the radius.

UNIT 7Rational Functions

Rational Functions!!!•RATIONAL FUNCTION: A function

which can be written as a fraction with 2 polynomial functions.

•Rules for simplifying/multiplying polynomials:

1. Multiply (if possible)2. Take out a GCF (if possible)3. Factor (if possible)4. Cancel out where possible5. Use exponent rules to reduce

Graphing Rational Functions

Vertical Asymptote: a vertical line that a rational function approaches but can never actually cross• Expressed as x = “?” lines• We find vertical asymptotes by

finding the values where the denominator of a rational function equals ZERO.

Graphing Rational Functions

Horizontal Asymptote: a horizontal line that a rational function approaches but can never actually cross• Expressed as y = “?” lines

Finding Horizontal Asymptotes

1. If the numerator has a lower highest power then the denominator, the HA is at the line y = 0.

2. If the numerator and the denominator have the same highest power, divide the leading coefficients to find the HA.

3. If the numerator has a higher highest power than the denominator, then the function has no horizontal asymptote.

X and Y-Intercepts!!!• To find the y-intercept, plug in “0” for

“x”.

• To find the x-intercept, set the numerator of the fraction equal to “0” and

INVERSE VARIATION!!!

• “k” is a number called the constant of variation.

• In these equations, as “x” gets bigger, “y” gets smaller. LIKEWISE, when “y” gets bigger, “x” gets smaller.

• Put the equation into your calculator.

DIRECT VARIATION• In these equations, as “x” gets bigger,

“y” gets bigger. LIKEWISE, when “x” gets smaller, “y” gets smaller too.

• “k” is STILL called the constant of variation. The equation for direct variation is y = kx.

• Put the equation into your calculator.

Transformations!!!• Quadratic F(x)s: k• Polynomial F(x)s: k• Exponential F(x)s: • Radical F(x)s: • Absolute Value: • Rational F(x)s: