Important Things to Remember on the SOL

Transcription

1 Notes Important Things to Remember on the SOL Evaluating Expressions *To evaluate an expression, replace all of the variables in the given problem with the replacement values and use (order of operations) *Absolute value bars take the answer inside and make it. *Treat,, and 3 like parentheses when deciding what order to solve. Laws of Exponents *To multiply terms with the same base, exponents, the numbers out front - Example: 8x 2 2x 6 = 8 2 x 2+6 = 16x 8 *To raise a term to a power, the exponent to each thing on the inside, making sure to exponents. - Example: (4x 3 ) 2 = 4 2 x 3 2 = 16x 6 *To divide terms, exponents and the numbers out front. - Example: 25x8 y 9 = 25 5x 4 y 5 x8 4 y 9 1 = 5x 4 y 8 *Whenever you see a negative exponent, SWITCH where the term goes in the fraction. Once you switch, make the exponent. - Example: 4x 5 2x 12 = 8x 7 = 8 x 7 Polynomial Operations *To add/subtract polynomials, find terms that have the same letters attached and combine the coefficients (numbers out front) using adding/subtracting - Example: (4x 3 5x + 3) + (6x + 2x 3 ) = (4x 3 + 2x 3 ) + ( 5x + 6x) + 3 = 6x 3 + x + 3 *To multiply two binomials, use (First Outside, Inside Last) - Example: (2x 3)(5x + 6) = 10x x 15x 18 = 10x 2 3x 18 *To divide a trinomial by a binomial, the top, then cancel the matching factors. - Example: (x 2 + 2x 15) (x + 5) = (x + 5)(x 3) (x + 5) = (x 3) *To multiply a polynomial by a monomial, to each term on the inside. - Example: 3x 2 ( 4x + x 2 1) = 3x 2 ( 4x) + 3x 2 (x 2 ) + 3x 2 ( 1) = 12x 3 + 3x 4 3x 2

2 Factoring Polynomials *If a problem asks you to factor, always check for a (GCF) first! *If there is a number in front of the x 2 term after taking out a GCF, make sure you factor (Multiply the first number by the last number and proceed as normal) *When factoring, you re always looking for two numbers that to get the last number, but to the middle number. *Insert a if a term is missing in the middle. * The word means a polynomial cannot be factored. Simplifying Radicals *Make a factor tree for the number underneath the * = circle * 3 = circle * thing comes to the outside from each circle Solving Literal Equations *Use to get the desired letter by itself. Solving Quadratic Equations *Quadratic equations are of the form ax 2 + bx + c = 0. *The solutions, roots, or are always put in { }. *You can ALWAYS use the quadratic formula to solve, which is: *The solutions,, or zeros of the graph of a quadratic equation are where the graph crosses the -axis.

3 Solving Multistep Equations *Multistep equations have all one variable with no x 2 term. *Follow DCMAD to solve: D C M A D *If the variables disappear and you get DIFFERENT numbers on each side of =, write. *If the variables disappear and you get the SAME number on each side of =, write. Solving Systems of Equations *A system of equations is two equations of the form Ax + By = C. We are looking for the ordered pair (x, y) that we can plug in to BOTH equations and BOTH are true. *We learned three methods of solution: substitution, elimination, and graphing. 1) Substitution Get a variable by itself in one of the equations Plug the expression into the OPPOSITE equation Solve for the variable Plug your answer back into the equation to solve for the other letter 2) Elimination Stack the equations so the letters and numbers match See if you can eliminate one of the letters by adding or If you can t, multiply the equations by a number to get them to match Eliminate one of the variables using or subtracting 3) Graphing Graph both equations. Where the lines cross is the solution. If the lines DON T cross, it s. If you get the SAME line, it s.

4 Solving Multistep Inequalities *Inequalities use the following four symbols:,,, or. *Solve a multistep inequality like you re solving an equation - Use (our rule for solving equations) *Flip the way the inequality sign faces if you or both sides by a negative number. *A solution to an inequality is any value that you can plug in to the inequality and get a true statement Solving Systems of Inequalities *To solve a system of inequalities, graph the inequality as if it was in y = mx + b form. - Use a dotted line ---- for the symbols or - Use a solid line for the symbols or - To decide which side of the graph to shade in, plug in a point to see if you get a true statement. *As long as it s not on the line, is the easiest to use. - Complete the same steps for BOTH inequalities. Solutions are points where the shading of both graphs overlap. Finding Slope *To find the slope given points (x 1, y 1 ) and (x 2, y 2 ), use the formula: *The slope of a line using a graph can be remembered as (change in y over change in x) * Don t forget HOYVUX! HOYVUX stands for *If the equation is not in y = mx + b form and it asks you for slope, get y by itself FIRST before you determine the slope! *A negative slope of a graph looks like it s going from left to right, while A positive slope looks like it s going from left to right.

5 Writing the Equation of a Line in y = mx + b *The equation for slope-intercept form is where is slope and b is the. * To write the equation, determine your (slope) first, then use the given point (x, y) to solve for (y-intercept) * Your final answer should only have numbers replacing the and the. Parallel and Perpendicular Lines *Parallel lines have the slope. *Perpendicular lines have slope (Flip and Switch) *Regardless of whether or not you have to change the slope, you always need to solve to find a new (y-intercept). When is a Relation a Function? *A relation is a function if all of the values are unique. *The graph of a function passes the line test. *Function notation is written as f(x), although they may use different letters than f. Domain and Range *The domain of a function is the values (the inputs) *The range of a function is the values (the outputs) *The domain of a graph contains all of the x (horizontal) values - If the graph is a line rather than individual points, the domain is smallest x-value x biggest x-value *The range of a graph contains all of the y (vertical) values - If the graph is a line rather than individual points, the range is smallest y-value y biggest y-value *To find the range of f(x) given a domain, plug in each value and use to solve. The range is each one of your answers put into { }

6 Zeros of a Function *The zeros of a function are values you can plug in and get as an answer. -DO NOT JUST PLUG IN ZERO! Zero is the answer you should get, not the input! *The zero of a graph is where the graph crosses the -axis. *Zeros are also called or. Intercepts *On a graph, the x-intercept is where a graph crosses the -axis - it is represented by the point (x, 0) * On a graph, the y-intercept is where a graph crosses the -axis - it is represented by the point (0, y) *If the problem gives you the equation in standard form and asks for the intercepts, use (but don t smack the paper!) Direct Variation * When y varies directly as x means to use the formula, where a is called the constant of. *Every graph of direct variation has two things in common: 1) It s a line (not curvy) 2) It passes through (also called the ) Inverse Variation * When y varies inversely as x means to use the formula, where k is called the constant of.

7 Line of Best Fit *If a question asks you for the line of best fit, it wants you to find the equation that BEST fits the points on the graph or in the table. IT WON T BE EXACT! *If the question asks for a line of best fit, the answer will be in y = mx + b form. - We can actually find this on the calculator. I will give you a sheet for this. *If the question asks for a curve of best fit, the answer will be in y = ax 2 + bx + c form. - We can actually find this on the calculator too. I will give you a sheet for this. *You can use these equations to help you predict other values by plugging in the given values into the equation and using PEMDAS to solve. Box-and-Whisker Plots *Range = - *Interquartile range = - Standard Deviation, Mean Absolute Deviation, and Z-Scores *Variance, standard deviation, and mean absolute deviation describe the data values relationship to the (average) *The bigger the standard deviation, the away from the mean the data is. *If the data has outliers, mean absolute deviation is a better measure than standard deviation or variance. * A z-score tells you how many a data point is away from the mean. (Positive means the mean, Negative means )

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