algebra 2 honors midterm review

Statistical question

A question that can be answered by collecting many pieces of information, or data, and summarizing that data.

Statistical variable

A quantity or quality for which data are expected to differ/that can be measured or counted, and for which data are expected to differ from one observation to another.

Categorical variable

A statistical variable for which the possible values belong to a limited set of qualitative responses/falls into categories (qualitative rather than quantitative attribute)

Quantitative variable

A statistical variable for which the possible values are numbers that can be meaningfully counted, added, subtracted, and so on/numerical data that can be compared, added, subtracted, or otherwise operated on.

Population

All the members of a set/the whole

Sample

A subset from the population is a subset of (some of) the population

Parameter

A piece of data from a whole population, not from a sample

Statistic

A piece of data from a sample, not a whole population.

Standard deviation

A measure of how much the values in a data set vary, or deviate, from the mean, x (line above).

Skewed Distribution

A distribution whose shape is stretched out in either the positive or negative direction (use median and quartiles to describe the center and spread of the data set)

Symmetric Distribution

A distribution whose shape is evenly distributed around the mean (use mean and standard deviation to describe the center and spread)

Normal Distribution

Shows data that vary randomly from the mean in the pattern of a bell-shaped curve

Mean

The sum of the data values divided by the number of data values

Median

The middle value in a data set. If the data set contains an even number of values, this is the mean of the two middle values.

Quartiles (Q1 and Q3)

The first and third are the medians of the lower half and upper half of the data, respectively.

Interquartile range

The difference between the third and first quartiles in a set of data

Frequency

Number of times an event or a value occurs

Frequency table

A table that lists items and shows the number of times the items occur. Consists of: class limit, frequency, and class boundary

Histogram

A diagram consisting of rectangles whose area is proportional to the frequency of a variable and whose width is equal to the class boundaries

Class limit

An interval for a group of data

Class boundary

The data values that separate classes. You find it by subtracting .5 from the lower limits and adding .5 to the upper limits

To find class limits

We need to know the class width

To find class width

max-min/# of classes (round or if it's a whole number +1)

Percentile

The percentage of values less than or equal to a particular data value. It is equal to the percentage of the total area under the distribution curve for the population to the left of that value.

About 68% of all values fall within

1 standard deviation from the mean

About 95% of all values fall within

2 standard deviations from the mean

About 99.7% of all values fall within

3 standard deviations from the mean

The Empirical Rule only applies to

Normal distributions

Parabola

The graph of a quadratic function.

Y-intercept

The point of the parabola on the y-axis, which occurs when x = 0.

X-intercepts

The points on the parabola on the x-axis, which occur when y = 0

Vertex

The lowest (minimum) or highest (maximum) point on the graph of the parabola

The axis of symmetry

The vertical line that divides the parabola into two congruent halves. It always passes through the vertex of the parabola. The x-coordinate of the vertex is the equation of this of the parabola

Quadratic function

A function with an equation that can be written in the form f(x) = ax2 + bx + c, with a =/ 0. All of these are transformations of the parent function defined by f(x) = x².

Vertex form

f(x) = a(x − h) 2 + k where (h, k) is the vertex of the parabola

The value of a determines

If -a; reflection over x-axis If |a|> 1: vertical stretch/ graph gets skinnier If 0<|a|<1: vertical compression/graph gets wider

The value of h determines

Horizontal shift in opposite direction/ going left or right

The value of k determines

Vertical shift in same direction/ up or down g(x)=a(x-h)²+k

A zero of a function

An x-intercept of the graph of a function

Zero Product Property

If ab = 0, then you can set a = 0 and b = 0

Perfect Square Trinomial

A trinomial is a perfect square trinomial if it can be factored into a binomial multiplied to itself. A perfect square trinomial factors into (x + b)²

Completing the Square

A method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial.

Quadratic Equations can be solved using the

Square Root Property by manipulating the equation until one side is a perfect square (completing the square).

When determining a function is increasing, decreasing, or constant we

Get the notation. interval notation and looking at x-values, but we will never have a closed bracket, always open parenthesis.

i

√−1

i² and i to the 6th power

-1

i cubed and i to the 7th and 8th power

-i

i to the 4th power

1

i to the 5th and 9th power

i

To find higher imaginary number's degrees we

Divide that number by 4

Imaginary unit (i)

The complex number whose square root is - 1: √−1

Imaginary number

An imaginary number is any number of the form bi , where b is a nonzero real number and i is the √−1.

Complex numbers

Numbers that can be written in the form a + bi , where a and b are real numbers and i is the square root of − 1 .

Complex Conjugates

Complex numbers with equivalent real parts and opposite imaginary parts

Factoring Difference of Squares

a²-b²=(a+b)(a-b)

Quadratic formula

x = -b ± √(b² - 4ac)/2a

Standard formula/quadratic equation

ax²+bx+c=0

Discriminant of a quadratic equation

b²-4ac

D<0

No real roots; two imaginary roots

D=0

One real root

D>0

Two distinct real roots

Monomial

Either a real number, a variable, or a product of real numbers and variables with whole-number exponent

Polynomial

A monomial or the sum or difference of two or more monomials.

Polynomial Function

A function whose rule is a polynomial

Commutative property

Order does not matter. Only multiplication and addition are commutative

Associative Property

The grouping of two or more numbers and performing the basic operations of multiplication and addition does not affect the final result

Distributive Property

The product of a single term and a sum or difference of two or more terms inside parentheses is the same as multiplying each addend by the single term and then adding or subtracting products.

Degree of a Polynomial

The largest exponent on one of the variables

Coefficient

The number/constant that's being multiplied into a variable

Leading Coefficient

In a polynomial, the coefficient that is attached to the highest degree variable

End behavior

Describes how the function is behaving at the ends of the function (as it approaches negative infinity and as it approaches positive infinity). It is affected by degree and leading coefficient.

Multiplicity

The number of times a given factor appears in the factored form of the equation of a polynomial

Even degree and Positive Leading Coefficient

End behavior: up, up

Even degree and Negative Leading Coefficient

End behavior: down, down

Odd degree and Positive Leading Coefficient

End behavior: falls left, rises right

Odd degree and Negative Leading Coefficient

End behavior: rises left, falls right

synonymous, the same as

solution, x-intercept, zero, root

Irrational Numbers

Repeating infinite decimals that can't be expressed as a fraction and/or square roots that are not perfect

Integer

Is a whole number

Rational numbers

Numbers that can be made by dividing integers by integers (as long as the denominator is not zero)

p

Constant

q

Leading coefficient

Rational Root Theorem

If p/q is a rational root of a polynomial equation, then p is a factor of the constant term and q is a factor of the leading coefficient. In other words: factors of p/factors of q

When adding or subtracting rational expressions with the same denominators

Add or subtract the numerators and combine like terms. Then simplify if you can.

When adding or subtracting rational expressions with unlike denominators

1. Factor each denominator. 2. Find the least common denominator (LCD) 3. Find the new numerator for each fraction. To find the new numerators for each fraction, compare the denominator of each of the original fractions to the LCD and write down everything different about the LCD in the numerator of the fraction. 4. Simplify the numerator by distributing first and then combining like terms. Combine the fraction by adding or subtracting the numerators. 5. Simplify the rational expression if you can. (i.e. Rewrite to identify factors in order to simplify.)

Property: Multiplication of Rational Expressions

Let p, q, r and s represent polynomials, such that q=/0, s=/0. Then p/q X r/s= pr/qs

Procedure: Multiplying Rational Expressions

1. Factor the numerators and denominators of all rational expressions. 2. Simplify the ratios of common factors to 1. 3. Multiply the remaining factors in the numerator, and multiply the remaining factors in the denominator.

Property; Division of Rational Expressions

Let p, q, r and s represent polynomials, such that q=/0, r=/0, s=/0. Then p/q ÷ r/s= p/q X q/r= ps/qr

Rational expression

The quotient of two polynomials

Simplified Form of a Rational Expression

A rational expression is in simplified form if its numerator and denominator are polynomials that have no common divisor other than 1.

Rational Equation

An equation that contains a rational expression

Extraneous Solutions

A solution of an equation derived from an original equation, but it is not a solution of the original equation

To solve rational equations

1. Multiply both sides of the equation by the common denominator to eliminate the fractions. 2. Simplify the resulting equation. To simplify the equation, you may need to distribute, or FOIL, and combine like terms. 3. Then solve the simplified equation for the variable. You can solve the equation by getting it equal to zero and factoring. 4. Confirm that the solution is valid in the original equation.

Asymptote

A line that a graph approaches. They guide the end behavior of a function

Vertical asymptotes

Vertical lines that correspond to the zeros of the denominator of a rational function, can occur at the x-values where the function is undefined

Horizontal asymptotes

Horizontal lines the graph approaches, can be crossed.

Case 1 of identifying horizontal asymptotes/The degree of the numerator is less than the degree of the denominator

As the value of x increases, the value of the denominator gets very large in relation to the numerator. The value of the function gets closer and closer to 0.

When the degree of the numerator is less than the degree of the denominator, there exists a horizontal asymptote at y = 0.

Case 2 of identifying horizontal asymptotes/The degree of the numerator is greater than the degree of the denominator

As the value of x increases, the value of the numerator gets very large in relation to the denominator. The value of the function continues to increase.

When the degree of the numerator is greater than the degree of the denominator, there are NO horizontal asymptotes.

Case 3 of identifying horizontal asymptotes

The degree of the numerator and the denominator are the same.

When the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.

FAIR

Factor (Find discontinuity if applicable)
Asymptotes
Intercept
Regions