Unit 1 and Unit 2 Concept Overview

Transcription

1 Unit 1 and Unit 2 Concept Overview Unit 1 Do you recognize your basic parent functions? Transformations a. Inside Parameters i. Horizontal ii. Shift (do the opposite of what feels right) 1. f(x+h)=left 2. f(x-h)=right iii. Reflect 1. f(-x)=reflect y-axis iv. Expansion/Compression 1. f(ax)=e/c horizontally 2. Do the opposite of what feels right a. a>1 compress b. a<1 stretch b. Outside Parameters i. Vertical ii. Shift (do what feels right) 1. f(x)+h=up 2. f(x)-h=down iii. Reflect 1. -f(x)=reflect x-axis iv. Expansion/Compression 1. af(x)=e/c vertically 2. Assume what feels right a. a>1 stretch b. a<1 compress Solving function values a. Take the value inside parameters (the parenthesis) and plug in the value every time you see your variable. b. g(x) = 3x + 7 c. g(4) = 3(4) + 7 Key Features a. Domain i. Values of x that the function can have ii. This is NOT your x-intercept iii. Perform the vertical line test. 1. Draw vertical lines across your function, if your line connects with the graph, then the domain must exist here 2. Find your leftmost vertical line to your rightmost 3. Draw logical conclusions about infinity b. Range i. Values of y that your function can have ii. This is NOT your y-intercept iii. Perform the horizontal line test. 1. Draw horizontal lines across your function, if your line connects with the graph, then the range must exist here 2. Find your lowest horizontal line to your highest

2 3. Draw logical conclusions about infinity c. Increasing/Decreasing i. An interval will increase as you move on the graph to the right, if the graph goes up ii. An interval will decrease as you move on the graph to the right, if the graph goes down d. Intercepts i. Y-intercept: set x = 0 and solve for y ii. X-intercept: set y = 0 and solve for x Testing for Symmetry a. Given a function, replace your y variable with -y respect to the x-axis b. Given a function, replace your x variable with -x respect to the y-axis c. Given a function, replace your y variable with -y AND replace your x variable with -x respect to the origin Testing Even/Odd/Neither a. Given a function, replace your x variable with -x i. If the signs simplifies to the original equation, this function is EVEN b. Given a function, replace your x variable with -x i. If the signs simplifies to the EXACT opposite signs of the original equation, this function is ODD c. Given a function, replace your x variable with -x i. If the signs simplifies to something other than the original equation or the EXACT opposite sign equation, this function is classified as NEITHER even or odd. How to Solve Compositions a. Blob Method i. First function = BLOB ii. Second function = PARENTHESIS iii. In your first function, find every variable and place a blob over the variable. ONLY the variable, leave everything else alone. iv. Take your entire equation for the second function and place it inside parenthesis v. Now, looking at your first function, begin copying everything down. vi. As you get to a blob, replace that with your parenthesis second function. vii. Continue until you have copied the entire NEW equation viii. Simplify ix. TADA! Composition Real Worlds a. Create at least 2 equations from the word problem b. Using your two equations, perform a composition i. Be sure to check both ways. What makes sense? c. Plug in your word problem information and solve d. READING your composition correctly i. f[g(x)] OR [f g](x) is read as g applied first e. STILL STRUGGLING WITH WORD PROBLEMS??? i. Solving Inverses a. What is an inverse?

3 Unit 2 i. Can pass horizontal line test? ii. It must be symmetric with y = x b. Steps for solving: i. Algebraic 1. Switch your variables x and y. 2. Solve for your new y 3. SOLVE means get the variable by itself! 4. f 1 ii. Graphical 1. Make a and switch your coordinate points (switch your x and y values) 2. Sketch new graph 3. Is it symmetric with respect to y = x? Solving Continuity a. Test the limit from the left b. Test the limit from the right c. Does the limit exist? i. Left = Right d. Test the function value e. Does the limit equal the function? Writing Limits/End Behavior a. Fill in the blanks to write a limit b. lim f(x) = x c. End Behavior i. If you know Domain and Range, you know your end behavior ii. Domain = (a, b); Range = (c, d) iii. Potential end behavior: 1. lim f(x) = c ; x a 2. lim f(x) = d x b Power Functions a. Any function written with an exponent AS LONG AS the variable is NOT in the exponent i. IF the variable is in the exponent, this is called an EXPONENTIAL function (which we learned about in UNIT 3) b. Are Power Functions i. x 2 ii. x 2 iii. x 1 2 iv. x 3 v. x c. Are NOT Power Functions i. 2 x ii. 5x 3 + 3x 2 1. This isn t a POWER because it has multiple terms. This makes it a POLYNOMIAL function Negative Exponents a. Wherever your negative exponent exists (in numerator or denominator), take the exponent AND it s base (ignore everything else), and flip it (move it from the bottom to the top, or vice versa)

4 b. Drop the negative c. Keep everything else the same and simplify d. Remember, nothing else move except the base and it s exponent Rational Exponents a. These are exponents that look like fractions b. Formula i. x A B B = x A c. Steps i. Rewrite your variable ii. Take the numerator and make it your new exponent 1. If your numerator is 1, ignore this step iii. Place a root around your base and exponent iv. Take the denominator and make it your root s number 1. If your denominator is 2, ignore this step Radical Functions a. Radical means Roots b. Converting your fraction exponents to radicals will help you solve them Solving Radicals a. Isolate the radical (only radical on one side of equals, everything else to other side of equals) b. Get rid of the radical (a square cancels a square root, a cube cancels a cube root, etc ) c. Solve for variable d. Check for EXTRANEOUS solutions (this means plug your answers back into the original equation and make sure everything makes sense) Polynomial Behavior a. A polynomial is a power function with multiple terms b. If n is the highest degree of a polynomial i. Then Possible Rational Zeros = n ii. Then Possible Turning Points = n 1 c. Leading Term Test i. A leading term is the coefficient of the term with the highest exponent ii. If leading term is positive 1. Degree Even U Up 2. Degree Odd - Opposite Arms going Up iii. If leading term is negative 1. Degree Even U Down 2. Degree Odd Opposite Arms going Down Solving Long Division a. Please see cheat sheet for visual solve Solving Synthetic Division a. Please see cheat sheet for visual solve b. Your Divisor should be a BINOMIAL c. To get your number on the outside of the box, SOLVE your DIVISOR d. Your polynomial Dividend should be in order from greatest exponent to smallest e. Bring all coefficients down (don t forget to bring it s sign with it) f. Bring the first coefficient down, multiply by number on outside, place that number in next empty spot. Add down g. Continue h. If zero is last number, it is even division i. If non-zero is last number, this is remainder

5 Finding Zeros of Polynomials a. Find possible rational zeros i. Theorem Factors of LAST term ii. Factors of FIRST term b. Use synthetic division until your get to quadratic depressed polynomial c. Factor d. Solve e. Check against possible rational zeros Solving Polynomial Inequalities a. Get one side to be zero i. If you DIVIDE or MULTIPLY by a negative number, please change the direction of your inequality b. Factor your quadratic (get it in the form (x ± a)(x ± b) c. Identify your roots (set equal to zero and solve) d. HANDY DANDY sign chart i. Place roots on number line e. Test your intervals f. Solve CAN YOU READ WORD PROBLEMS?!?!?!?!?!?!