CH.P Pre-Calculus Review

CH.P Pre-Calculus Review BOWTIE METHOD Bowtie Method is a shortcut method to add and subtract fractions. 25 7-2-3-5 29 37 3-7 21 EX#1: 14 9-20-11 59 45 = 45 EX: 5 х 4-5-20- x4 4 4x EX#2: sinxcosx sinx+(メ COSX SIX COSX Six EX#3: 363 xx+4 x(x+4) 4x+12 XLX-4) x(x-4) FACTORING EX#1: Factor and solve each. INFORMATION ABOUT LINES: To write an equation of a line you need: 1) Slope and 2) A point Slope formula: ****Point-slope form: y-y=m(x-x) (Best Form to Write) **Slope-intercept form: y=mx+b (m=slope and by-intercept) (Best Form to Read) Standard form: Ar+By=C (A must be positive and A, B, and C must be integers) General form: Ax+By+C=0 (A must be positive and A, B, and C must be integers) Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes of each other. (Ifm=5, then m₁=3) 5 WRITING EQUATIONS OF LINES EX: Write the equation of the line that passes through (2,-3) and (5.7). 用= 7-(-3) 10 5-2 3 10 10 Point-slope form: 3+3=(x-2) or y-7=(x-5) You can use either point. 3 10-29 Slope-intercept form: y=x 3 Standard form: 10x-3y=29 General form: 10x-3y-29-0 EX#1: Write the equation of each line a) (4,-7); m=11 Y+7=11 (x-4) b) (-5,-8) (11,-10) m= 1/6 +8= (x+5) 8 Y+ 10=(x-11) EX#2: Write the equation of the line that is..... a) parallel to the line y=3x+8 and passes thru the point (-6,4). 4-4=-3(x+6) b) 1 to the line 4x-5y=7 and passes thru the point (9,-1). 4+1=(x-9) EVEN AND ODD FUNCTIONS A function is even iff it is symmetric about the y-axis. If f(x)=f(-x) then the graph is symmetric about the y-axis. (Even Function) EX: f(x)=x+4x and g(x)=7-x² are both even functions because all the powers are even. y=cor x is an even function because it is symmetric about the y-axis. A function is odd iff it is symmetric about the origin. If f(x)=f(-x) then the graph is symmetric about the origin. (Odd Function) EX: f(x)=x+5x and g(x)=7x-x are both odd functions because all the powers are odd. y=sinx is an odd function because it is symmetric about the origin. A function is neither even or odd if its' powers are mixed (some even and some odd). EX: f(x)=x²+5x and g(x)=7-x are neither even or odd functions because the powers are mixed. COMPOSITION OF FUNCTIONS: (f°g)(x) and (gf)(x) (f°g)(x)=f(g(x)) means plug g(x) intof(x) (gf)(x)=g(f(x)) means plugf(x) into g(x) (fog)(2)=f(g(2)) which means plug 2 into g(x) and that answer into f(x). (gf)(6)=g((6)) which means plug 6 into f(x) and that answer into g(x). DOMAIN DOMAIN: the x values that work in an equation. (How far left and right the graph goes) There are three things in math that you cannot do and they all will restrict your domain 1) You cannot divide by zero. 2) You cannot take the square root of a negative number. 3) You cannot take the logarithm of a negative # or zero. If there are no fractions, radicals, or logarithms, then the domain has no restrictions and therefore the domain is all real #'s. For fractions the denominator cannot = 0, so the # that makes the bottom = 0 is not in the domain. EX: f(x)= 3 x-11 エー8 EX: f(x)=x+3x-10 (x+5)(x-2) Domain: x=11 Domain: x-5,2 Since you cannot take the square root of a negative # the function inside the must be 20 EX: f(x)=√16-x 16-120 162x Domain: x≤16 Yes 0 16 No EX: f(x)=x+5 x+520 12-5 Domain: x2-5 No 0 Since you cannot take the square root of a negative # or divide by zero, you must find #'s that make the square root 20 but don't make the denominator = 0. (The best technique is to organize yourself with a number line) EX:( √2-x 10 EX: f(x)=-7 Domain: x2,x#1 or (-0, 1)(1, 2) yes U yes 0 по 2 Domain: x>7 No Yes 7 Since we cannot take the log/ In of a negative # or zero, the function that we are taking the log/In of must be > 0. EX: f(x)=ln(6-x) 6-x>0 6> Yes 24 No EX: f(x)=log(x+8) x+8>0 x>-8 No Domain: <6 Domain: -8 If there are no fractions, radicals or logarithms, the function has no restrictions and has a domain of all real #'s, EX: f(x)=x, g(x)=sinx, h(x)=cosx, (x)=3x-2, A(x)= Domain; all real #s 8 +5 RANGE: the y values that work in an equation. (How high and low the graph goes) INCREASING: as x gets larger the y values get larger. (The graph goes up to the right) DECREASING: asx gets larger the y values get smaller. (The graph goes down to the right)