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Open circle : greater than or less than
Closed circle : Greater than or equal to, or Less than or equal to

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Has a variable in the equation

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You need to remember the open and closed circles if it has a “equal to” than it is closed but if it doesn’t than it’s opean on the number line

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You need to remember the open and closed circles if it has a “equal to” than it is closed but if it doesn’t than it’s opean on the number line

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When graphing, one variable= number line

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Vertical line- undefined slope
Horizontal line- zero slope

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Greater then or equal too + less than or equal too are closed circles

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Greater then or equal too + less than or equal too are closed circles

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Greater than + less then are open circles

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Base equation: y=a|x-h|k

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you flip the inequality sign whenever you multiply or divide the inequality by a negative

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You can solve systems of equations through different methods: graphing, substitution, elimination

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Slope intercept form: y=mx+b
Standard form: ax+by=c
Point slope form: y-y1=m(x-x1)

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Shade down for less than and shade up for greater than

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Shade down for less than and shade up for greater than

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Caleb

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There are three ways to solve systems of linear equations in two variables: graphing. substitution method. elimination method.

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When you use elimination, eliminate a variable, multiply by that variable if necessary, solve, and plug the values in. Always write the solution as an ordered pair.

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Substitution method 1:
1. Set the equations equal to each other
2. Solve for the variable that remain
3. Plug in the value you found into one of the original equations
4. Solve for the remaining variable
5. Write your final answer as an ordered pair

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The solution is the part on the graph where the inequalities overlap

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Translating words into mathematical symbols in order to solve word problems using equations, how to solve equations and inequalities that contain absolute value symbols, inequalities, how to represent your solutions graphically on a number line. 

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Translating words into mathematical symbols in order to solve word problems using equations, how to solve equations and inequalities that contain absolute value symbols, inequalities, how to represent your solutions graphically on a number line. 

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The Parent Function for Absolute Value Equations

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The Parent Function for Absolute Value Equations

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The Parent Function for absolute value equations is y=a|x-h|k. If there is not a number for a variable, that’s ok as long as there is an x value. The variables are called transformations. These change the look of the graph. For example, h and k move them in a horizontal and vertical way, while a moves them in regards to the stretching or compressing of the graph.

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Make sure you know what the parent function is

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Flip it when you multiply by negative

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To solve a system of equations, if one of them has a variable by itself, like y=2x+3, you can plug that into the other equation,
6=2y+3x
Plug in 2x+3 as the thing that y equals (sorry I forgot how to english)
And solve

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The solution is an ordered pair

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Remember all the methods

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Write the answers in ordered pair

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Where the overlap is where the solution is and if they don’t there is no solution.

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Where the overlap is where the solution is and if they don’t there is no solution.

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Shade above when it's positive

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Inequalities contain two variables of at least two linear inequalities of the same variable.

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The intersection of the 2 lines is the ordered pair that works for the 2 equations

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The intersection of the 2 lines is the ordered pair that works for the 2 equations

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Remember when you are graphing to look at the equation symbol so you know if you need to shade above or below of your equation on the graph.

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Linear equations have equal signs while inequalities contain greater than and less than signs.

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To solve a subsitution problem you can solve one equation and then plug the answer into the other one to find the opposite variable so you have solutions to the variables of the equation.

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There will be no solution if the lines do not overlap

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The graph of a quadratic function is called a parabola then to solve it you have to find the x intercepts, y intercept, axis of symmetry, and the vertex or maximum or minimum value

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The first step in solving a polynomial is to find its degree and an example is: 4x +3 and it is linear because there is no exponent and it has a variable to make it linear and not constant

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Monomial has 1 term
Binomial has 2 terms
Trinomial has 3 terms
Polynomial has 4 or more terms

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The parent function for a quadratic function is y=x^2
Then you add all the other stuff to get y=ax^2+bx+c
Yeah, the graph is a parabola. When it opens upward, you have a minimum value, downward, it means you’re “a” value is negative, and you have a maximum value. These are useful in real life problems having to do with money.

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Solving quadratics:
There are four methods
Completing the square
Factoring
The quadratic formula
Graphing

😊

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To solve Polynomials by factoring, first make sure the equation in correct form. (This means putting the degrees in order.) Then set the equation equal to zero. Next, you factor the problem. (Which is the hardest part.) After that, you set the individual factors equal to zero, solve, and tada! One solve equation. Most often you will have three solutions. 📝😁

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To factor, you multiply the "A" and "C" values together, and then find two numbers that multiply to that (A x C) value that add up to "B". Plug those numbers in, take out the factors from the two pairs and finish by using one of the equal factored pairs and the two numbers factored out, set them equal to zero, and find your zeros!

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When solving/ graphing a polynomial, you can determine the maximum number of zeros simply by looking at the highest degree.

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Whenever A is negative, the graph has a Max
Whenever A is positive, the graph has a Min

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Quadratics have a degree of 2 that is what mekes them quadratic. You can solve them by completing the square, factoring, and the quadratic formula. The quadratic formula is -b+and-thesquare root of -b+4AC all over 2A.

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Quadratics have a degree of 2 that is what mekes them quadratic. You can solve them by completing the square, factoring, and the quadratic formula. The quadratic formula is -b+and-thesquare root of -b+4AC all over 2A.

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Quadratic equation: y=ax^2+by+c
When a is positive the parabola opens up and the vertex is a minimum. When a is negative the parabola opens downs and the vertex is a maximum.
Parent function: y=x^2

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When you have a quadratic function and a is positive the parabola opens upward and the vertex is a min but when the a is a negative number the parabola opens downward and the vertex is a max.

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To find the end behavior of a polynomial function on a graph you look at the quadrants on the graph and the put + or - infinity depending on where it's located on the graph. Ex.(L) x---> - infinity as y---> +infinity (R)x---> + infinity as y---> -infinity.

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Polynomial functions: quadratic, cubic, quartic and so on, involving only non negative integer powers of x

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Has a max and min, it can also have zeros

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Has a max and min, it can also have zeros

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All the odd degree polynomials behaviors are similar and same for the even degrees

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All the odd degree polynomials behaviors are similar and same for the even degrees

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Greater than is shaded inside the quadratic and less than is shaded outside the quadratic

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Y=ax^2+by+c is the Parent Function for a quadratic. Based on that, I’m going to tell you how I factor. I look at the a and c values, and multiply them together. Then I ask myself, “what factors of the sum of a and c will add to give me the b value? Once I’ve figured that out, those two factors are what go into my final answer and then it is factored.

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The vertical line through the X coordinate of vertex is the axis of symmetry

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If the hole is open you don't continue the line

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To solve you can complete the square, the quadratic formula, factor, or graph

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If the graph passes the x axis a the zero or touches the x axis and turns around the zero

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A monomial is 1 term, Ex: x
A binomial is 2 terms, Ex: x+1
A trinomial is 3 terms, Ex: x^2-2x+1
A polynomial is or more terms, Ex: x^3-2x^2+1x-1

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Before finding the solution to a polynomial. The polynomial must be in order according to the degree.
Ex: a^2 +a^8 +6 +c^3. ->.  a^8  + c^3 + a^2 +6

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When 'a' is positive, the parabola opens upward and there is a minimum value. When 'a' is negative, the parabola opens downward and there is a maximum value

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The turning point is where the graph changes direction

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Quadratic functions form is Ax^2+Bx+C

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A Value, K value and H value if A is negative it flips and compressed but if it’s positive it just compresses. If K is negative than it moves down but if it’s positive than it moves up. If H is negative it moves right but if it’s positive it moves left.

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A Value, K value and H value if A is negative it flips and compressed but if it’s positive it just compresses. If K is negative than it moves down but if it’s positive than it moves up. If H is negative it moves right but if it’s positive it moves left.

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A logarithm is the inverse of an exponent. 

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A logarithm is the inverse of an exponent. 

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Expanded Form
Log b (AC)= Log b (A)+ Log b (C)
Log b (A^d)= d Log b (A)
Log b (A/C)= Log b (A) - Log b (C)
Condensing is the opposite of expanded so just flip it around and that’s condensed. ( Left is condensed right is expanded)

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To find VA set the bottom equal to 0 and solve.

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To find VA set the bottom equal to 0 and solve.

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To find HA take the top power and bottom power (biggest one) Than is the top is bigger than there is no HA but there is a SA. If the bottom is bigger than it is equal to zero and if they are equal than you divide the top by the bottom (coefficient). If
there is a SA you find it by long division.

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When you first graph a radical function, you must consider the domain of the function. Like what numbers would represent that x value or variable.

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Graphing radical equations is probably the first time you've had to consider the domain of the equation before you graph. This is because you cannot put a negative value inside a square root

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There's multiple ways to find the functions and the graphs.

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An exponential function is a mathematical expression in which a variable represents the exponent of an expression.

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you can take the odd root of any  number positive or negative but can only take the even root of positive numbers.

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you're never ever allowed to divide by zero when solving a rational function.

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When solving a log equation you can sometimes put it into exponential form to find the answer.

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The first step when solving a rational function is to factor the numerator and denominator, then to find the x-intercept find the real zeros of the numerator and to find the y-intercept plug in zero for x. Then to find the VA find the real zeros of the denominator an or the HA compare the degrees of the numerator and denominator.

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use the Leading Coefficient Test to determine the graph's end behavior.

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Basic form is f(x)=b^2

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You cannot have a radical in the denominator

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You cannot have a radical in the denominator

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Never divide by 0 or by the square root of a negative

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Y=1/X

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A radical equation is an equation in which a variable is under a radical. To solve a radical equation: Isolate the radical expression involving the variable. If more than one radical expression involves the variable, then isolate one of them. Raise both sides of the equation to the index of the radical.

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To find the slant asymptote, use long division.
Z(x) / P(x)