MS400-Chapter1+2

graphs of functions

y=f(x), for each input value along the horizontal axis there is exactly one output value corresponding: LINE, WAVES

one-to-one graph or not

A Graph is not a one-to-one function if the output value has two corresponding input values. A Graph is a one-to-one function when each input corresponds to exactly one output

Determinant Formula

A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities.

Function

A rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value uniquely determines one output value. We say "the output is a function of the input."

Solve an equation involving a function

Solving equations involving a function is what we do when we know an output, and use the function to determine the inputs that would produce that output. Solving a function could produce more than one solution

one-to-one function.

Sometimes in a relationship each input corresponds to exactly one output, and every output corresponds to exactly one input.

Identity Function Graph

f(x) = x

Reciprocal Function

f(x)=1/x

Cube Root Function

f(x)=3√x

Quadratic

f(x)=x^2

Cubic Function

f(x)=x^3

Square root function

f(x)=√x

An independent system

has exactly one solution pair(𝑥,𝑦). (x,y). The point where the two lines intersect is the only solution. CROSSING LINES

A dependent system

has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations. ONE LINE

An inconsistent system

has no solution. Notice that the two lines are parallel and will never intersect. Parallel

The multiplicative inverse of a matrix is

similar in concept, except that the product of matrix𝐴A and its inverse𝐴−1 A−1 equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by𝐼𝑛 In where𝑛 n represents the dimension of the matrix.

break-even point.

The point at which the two lines intersect

Pearson's r can range from -1 to 1.

An r of -1 indicates a perfect negative linear relationship between variables, an r of 0 indicates no linear relationship between variables, and an r of 1 indicates a perfect positive linear relationship between variables

horizontal line test (mult Xs)

Determine if a function is a one-to-one function. If any horizontal line crosses the graph more than once, then the graph does not define a one-to-one function.

Evaluate a function

Evaluating a function is what we do when we know an input, and use the function to determine the corresponding output. Evaluating will always produce one result, since each input of a function corresponds to exactly one output.

Find the Determinant of a 3x3

Find the determinant of the 3×3 matrix. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal. |𝐴|=𝑎1𝑏2𝑐3+𝑏1𝑐2𝑎3+𝑐1𝑎2𝑏3−𝑎3𝑏2𝑐1−𝑏3𝑐2𝑎1−𝑐3𝑎2𝑏1| A |=a1b2c3+b1c2a3+c1a2b3−a3b2c1−b3c2a1−c3a2b1

How to represent functions

Functions can be represented in many ways: Words (as we did in the last few examples), tables of values, graphs, or formulas.

The vertical line test (mult Ys)

If any vertical line would cross the graph more than once, then the graph does not define only one vertical output for each horizontal input.

Solve a 2x2 matrix using Cramer's

If we are solving for𝑥, x, the𝑥 x column is replaced with the constant column. If we are solving for𝑦, y, the𝑦 y column is replaced with the constant column.

Gaussian elimination

Row-reduce until you get an x = number and substitute until you get an actual number. 𝐴𝑥+𝐵𝑦+𝐶𝑧=𝐷 𝐸𝑦+𝐹𝑧=𝐺 𝐻𝑧=𝐾

Gaussian elimination steps: Given an augmented matrix, perform row operations to achieve row-echelon form with example

The first equation should have a leading coefficient of 1. Interchange rows or multiply by a constant, if necessary. Use row operations to obtain zeros down the first column below the first entry of 1. Use row operations to obtain a 1 in row 2, column 2. Use row operations to obtain zeros down column 2, below the entry of 1. Use row operations to obtain a 1 in row 3, column 3. Continue this process for all rows until there is a 1 in every entry down the main diagonal and there are only zeros below. If any rows contain all zeros, place them at the bottom.

function notation

The notation output = f(input) defines a function named f. This would be read "output is f of input"

Solve a 3x3 matrix using Cramer's

Using matrices: 1st matrix is just the coefficients, the second third and fourth have the result replacing the coefficient you are solving for

matrix in augmented form

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However,

Can the substitution method be used to solve any linear system in two variables?

Yes, but the method works best if one of the equations contains a coefficient of 1 or -1 so that we do not have to deal with fractions.

Pearson's r

a correlation coefficient that designates the magnitude of a relationship between 2 variables

A system of linear equations

consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.

Find the Determinant of a 2x2

create a matrix, multiply the indices in the diagonal and subtract them from each other

Linear Constant

f (x) = c , where c is a constant (number)

Absolute Value

f (x) = | x |

Cramer's Rule description

is a method that uses determinants to solve systems of equations that have the same number of equations as variables.

A system of nonlinear equations

is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form𝐴𝑥+𝐵𝑦+𝐶=0. Any equation that cannot be written in this form in nonlinear.

Linear Transformation A linear transformation

is any transformation of a variable that can be achieved by multiplying it by a constant, and then adding a second constant. If Y is the transformed value of X, then Y = aX + b. The transformation from degrees Fahrenheit to degrees Centigrade is linear and is done using the formula:C = 0.55556F - 17.7778.

The most common criterion used to determine the best-fitting line

is the line that minimizes the sum of squared errors of prediction. This line does not need to go through any of the actual data points, and it can have a different number of points above it and below it.

The profit function

is the revenue function minus the cost function, written as𝑃(𝑥)=𝑅(𝑥)−𝐶(𝑥). P(x)=R(x)−C(x). Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

row-echelon form,

to solve the system of equations, we want to convert the matrix in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.

Cramer's Rule Formula

𝐷: D:determinant of the coefficient matrix 𝐷𝑥:Dx:determinant of the numerator in the solution of𝑥 x=𝐷𝑥/𝐷 𝐷𝑦:Dy:determinant of the numerator in the solution of𝑦 𝑦=𝐷𝑦/𝐷