Chapter R Notes only.pdf

R.1 Notes Rules and Definitions: The standard equation of a line is Ax+By=C where A, B, and C are real numbers, A & B not both 0. The x-intercept of a graph is where it crosses the x-axis it is found by letting y = 0 and solving for x. The y-intercept of a graph is where it crosses...

R.1 Notes Rules and Definitions: The standard equation of a line is Ax+By=C where A, B, and C are real numbers, A & B not both 0. The x-intercept of a graph is where it crosses the x-axis it is found by letting y = 0 and solving for x. The y-intercept of a graph is where it crosses the y-axis it is found by letting x = 0 and solving for y. 𝑦2 – 𝑦1 The slope of a line through (x1, y1) and (x2, y2) is m = this is referred to as Rise over Run. 𝑥2 −𝑥1 A vertical line has an equation of x = some real number, its slope is undefined. A horizontal line has an equation of y = some real number, its slope is zero. The slope-intercept equation of a line is y = mx +b, where m = slope and (0, b) is the y-intercept The point-slope equation of a line is 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) where m = slope and (x1, y1) is a point on the line. Given two lines L1 and L2, with slopes m1 and m2 respectively 1. L1 and L2 are parallel if and only if m1 = m2 (Slopes are the same) 1 2. L1 and L2 are perpendicular if and only if 𝑚1 = − 𝑚 (Slopes are opposite reciprocals) 2 Order of Operations (PEMDAS) 1. P – Parentheses or other grouping symbols (fraction bars, absolute values, brackets, etc.) 2. E – Exponents (powers and radicals) 3. MD – Multiplication and Division done in order from left to right 4. AS – Addition and Subtraction done in order from left to right Exponent Rules 𝑎 𝑛 𝑎𝑛 1. 𝑎0 = 1 and 𝑎1 = 𝑎 6. (𝑏 ) = 𝑏𝑛 1 2. 𝑎𝑚 𝑎𝑛 = 𝑎𝑚+𝑛 7. 𝑎 −𝑛 = 𝑎𝑛 𝑎𝑚 1 3. = 𝑎𝑚−𝑛 8. = 𝑎𝑛 𝑎𝑛 𝑎 −𝑛 𝑎 −𝑛 𝑏 𝑛 4. (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 9. (𝑏 ) = (𝑎 ) 𝑚 𝑛 𝑛 5. (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏𝑛 10. 𝑎 𝑛 = √𝑎𝑚 = ( √𝑎 )𝑚 When solving an inequality (, ) if you multiply or divide by a negative number you must flip the inequality (For example if you had < after multiplying by a negative you would have > ) You need to know interval notation. If you do not recognize the following please talk to me! 1 1 For example, 𝑥 < −3 is (−∞, −3), < 𝑥 ≤ 5 is ( , 5], and 𝑥 ≥ 0.63 is [0.63, ∞) 2 2 A function is a relation in which every x value corresponds with exactly one y value. The domain is the set of all x values for which the function results in a real number, the range is the set of all resulting y values. Function Notation: If an equation is a function, we typically set y = 𝑓 (𝑥 ) (it can be any letter, it does not have to be𝑓). For example the line, 𝑦 = 3𝑥 − 5 could also be written 𝑓(𝑥) = 3𝑥 − 5. For 𝑦 = 3𝑥 − 5, if x=2 then y = 1 and the point (2, 1) is on the line, using our function notation we would have 𝑓 (2) = 1 it also means that the point (2, 1) is on the graph. Vertical Line Test: A graph is a function if and only if no vertical line intersects the graph more than once. Finding the domain algebraically: 1. Make sure values under a square root (or even root) are greater than or equal to zero 2. Do not include an x value that will make the denominator of a fraction equal to zero R.2 Notes Rules and Definitions: Steps in Factoring: 1. Check to see if there is a GCF (greatest common factor), if so factor it out. 2. Count the number of terms If there are 4 terms: o Try grouping the first 2 terms and the second 2 terms and factoring out a GCF from both groupings. If there are 3 terms: o 𝑥 2 + 𝑏𝑥 + 𝑐, use (𝑥 )(𝑥 ) where the number multiply to c and add to b o 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, Find two numbers that add to be b and multiply to ac, rewrite bx using these two numbers then use the process of grouping. If there are 2 terms: o 𝑥 2 − 𝑎2 = (𝑥 − 𝑎)(𝑥 + 𝑎) o 𝑥 2 + 𝑎2 this is prime, it does not factor o 𝑥 3 − 𝑎3 = (𝑥 − 𝑎)(𝑥 2 + 𝑎𝑥 + 𝑎2 ) o 𝑥 3 + 𝑎3 = (𝑥 + 𝑎)(𝑥 2 − 𝑎𝑥 + 𝑎2 ) 3. Look at your answer, if any of the factors factor further, do so Zero Product Property: If ab=0 then a=0 or b=0. This allows us to solve: (𝑥 − 1)(3𝑥 + 2) = 0 by setting 𝑥 − 1 = 0 𝑎𝑛𝑑 3𝑥 + 2 = 0 and solving each equation. The imaginary number: 𝑖 = √−1 𝑎𝑛𝑑 𝑖 2 = −1 A complex number has the form 𝑎 + 𝑏𝑖 where 𝑎, 𝑏 are real numbers 𝑎 + 𝑏𝑖 and 𝑎 − 𝑏𝑖 are complex conjugates, their product is always a positive real number. Solving with the Square Root Method 1. Get the squared term on its own side of the equation 2. Take the square root of both sides, remember the side with a number gets a ± 3. Solve for the variable if needed −𝑏±√𝑏2 −4𝑎𝑐 Quadratic Formula: The solution(s) to 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 is given by 𝑥 = 2𝑎 R.3 Notes Ex. Rewrite without radicals and fractions 2 4 1 A. B. √𝑥 3 C. 𝑥5 √𝑥 Ex. Simplify 2 5 𝑥−9 A. B. C. √5 √11−1 √𝑥+3 2 1 9 + E. 3 5 D. 2 1 √𝑥+3 – 3 6 1 1 1 −2 − 2−1 G. 𝑥 2+ℎ F. 1 𝑥 – ℎ 3 6 R.4 Notes (Page 1) Eight Basic Graphs: 𝑦=𝑥 𝑦 = 𝑥2 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 y = x3 𝑦 = √𝑥 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 3 𝑦 = √𝑥 𝑦 = |𝑥| 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 R.4 Notes (Page 2) 1 1 𝑦=𝑥 𝑦 = 𝑥2 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 How can we “mess” with our graphs? 1. 2. 3. 4. In general…. things that change y-values are on the “outside” of the function and things that change x- values are on the “inside” of the function. Ex. Graph the following, label at least 2 points on each graph. A. 𝑦 = √𝑥 − 2 B. 𝑦 = |𝑥 − 2| 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 R.4 Notes (Page 3) C. 𝑦 = −𝑥 2 D. 𝑦 = √−𝑥 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 1 E. 𝑦 = 2|𝑥| F. 𝑦 = 3 𝑥 3 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 1 G. 𝑓(𝑥 ) = −3|𝑥 + 1| − 2 H. ℎ(𝑥 ) = (𝑥−2)2 + 1 8 8 6 6 4 4 2 2 5 5 5 5 2 2 4 4 6 6 8 8 R.5 Notes (Page 1) Def. For all values for which both 𝑓 (𝑥 ) and 𝑔(𝑥 ) are defined 1. (𝑓 + 𝑔)(𝑥 ) = 𝑓(𝑥 ) + 𝑔(𝑥 ) 2. (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥 ) − 𝑔(𝑥 ) 3. (𝑓 ∙ 𝑔)(𝑥 ) = 𝑓 (𝑥 ) ∙ 𝑔(𝑥 ) 𝑓 𝑓(𝑥) 4. ( ) (𝑥 ) = ℎ𝑒𝑟𝑒 𝑔(𝑥 ) ≠ 0 𝑔 𝑔(𝑥) Note: Domain of 1 through 4 are the common elements of the domains of 𝑓 (𝑥 ) and 𝑔(𝑥 ). 1 Ex. Given 𝑓 (𝑥) = 𝑥 2 − 1 and 𝑔(𝑥 ) = 𝑥 3 find (𝑓 ∙ 𝑔)(𝑥) Ex. Given 𝑓 (𝑥) = 3𝑥 − 1 and 𝑔(𝑥 ) = √𝑥 find the domain of (𝑓 − 𝑔)(𝑥 ) 𝑓(𝑥+ℎ)−𝑓(𝑥) Def. The Difference Quotient for a function 𝑓(𝑥 ) is: ℎ Picture: R.5 Notes (Page 2) Ex. Find the difference quotient for 𝑓 (𝑥 ) = 3𝑥 2 − 1 Def. Let 𝑓 and 𝑔 be functions, if 𝑔(𝑥 ) is in the domain of 𝑓(𝑥 ) then 𝑓 ∘ 𝑔 is the composition of 𝑓 and 𝑔. (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥 )) Note: The domain of 𝑓 ∘ 𝑔 are the common elements of the domain of 𝑔(𝑥 ) and the domain of the resulting equation when finding 𝑓 ∘ 𝑔 1 Ex. Given 𝑓 (𝑥) = and 𝑔(𝑥 ) = √𝑥 − 1 find (𝑓 ∘ 𝑔)(𝑥) and its domain. 𝑥2 R.6 Notes Ex. Solve for x. 𝑥+2 𝑥+2 𝑥 12𝑥 𝑥−1 A. + =2 B. − = 𝑥+1 𝑥+4 𝑥−4 𝑥 2 +𝑥−20 𝑥+5 C. 𝑥 + 3 = √𝑥 + 5 D. √3𝑥 + 13 + 3 = 2𝑥 R.7 Notes Ex. Solve A. |𝑥| = 5 B. |𝑥 + 1| = 7 C. 2|3𝑥 − 1| = 10 Ex. Solve. Graph and give the solution in interval notation. A. |𝑥| ≤ 2 B. |𝑥| > 4 C. |1 − 2𝑥| ≤ 5 D. |7 + 3𝑥| ≤ −1 E. |7 − 3𝑥| + 4 ≥ 0