8 Week Exam

lim x --> #

# is a vertical asymptote

Exponential graphs always have

1 HA 2 points

Logarithmic graphs always have

1 VA 2 points

State how many complex and real zeros the function has: f(x) = x^2 -10x +41

1. # of complex zeros = degree = 2 2. # of real zeros = plug into graphing calc = 0 3. Sketch graph with viewing window

Sketch the polynomial function graph for the given zeros and multiplicities -4 multiplicity of 3 3 multiplicity of 2

1. Convert zero to factor form (x+4)^3 (x-3)^2 2. Odd multiplicities = cross x axis 3. Even multiplicites = touch x axis

Find a polynomial with real coefficents that has the given zeros: -1, 5-4i

1. Convert zeros to factors (x + 1) (x - (5 - 4i) ) (x - (5 + 4i) ) (X + 1) (x -5 +4i) (x -5 -4i) 2. Multiply and Simplify x^3-9x^2+31x+41

Write a polynomial function minimum degree in standard form with real coefficents whose zeros include those listed zeros: 3i and -3i

1. Convert zeros to factors (x - 3i) (x + 3i) 2. Multiply and Simplify x^2 + 9

Find the exact solution algebriacally 2 * 2^x/3 = 32

1. Divide 2^x/3 = 16 2. Find out what 2 raised to = 16 2^4 = 16 2^x/3 = 2^4 x/3 = 4 x = 12 {12}

Graph (calculator) the polynomial function, and locate its extrema and zeros: f(x) = -x^4 + 13x

1. Draw graph with viewing window 2. x = 0 x = cubrt(13) -- 2.351

Tell whether the function is an exponential growth or decay and find the constant rate of crowth /decay f(x) = 4.8 * 1.05^x

1. Find r 1.05 = 1 +r r = .05 2.Compare r r>0 = true so function is expoential growth

State the domain of the function of f(x) = log[x(x+7)]

1. Find the Vertical asymptote x= 0 x = -7 2. Place into a sign chart + -7 - 0 + 3. Find the x intercepts: x(x+7) = 1 x^2 + 7x -1 = 0 quadratic formula x = 0.140 x = -7.140 4. State the domain (-inf , -7) U (0, inf) 5. Choose two random points to the left and right, and then plot them and draw curves

How to solve polynomial inequalities

1. Find the Zeros and Undefined Values 2. Place Zeros on the sign chart 3. Test the "Regions"

Determine the real values of x that can cause the function to be a) Zero, b) Undefined, c) Positive, d) Negative f(x) = x sqrt(x+10)

1. Find the zeros (What makes the numerator 0) x = 0 sqrt(x+10) = 0 x = -10 2. Undefined (Negative roots, or 0 denominator) x < -10 3. Place values on sign chart and test regions Cross out anything behind -10 4. Determine what makes the function positive f(x) > 0: (0, inf) 5. Determine what makes the function negative f(x) < 0: (-10, 0)

Determine the real values of x that can cause the function to be a) Zero, b) Undefined, c) Positive, d) Negative f(x) = (x-1) / (5x+8)(x-6)

1. Find the zeros (What makes the numerator 0) x-1 = 0 x = 1 2. Undefined (Negative roots, or 0 denominator) 5x+8 = 0 x = -8/5 x-6=0 x=6 3. Place values on sign chart and test regions 4. Determine what makes the function positive f(x) > 0: (-8/5, 1)U(6, inf) 5. Determine what makes the function negative f(x) < 0: (-inf, -8/5)U(1,6)

Graph the rational function with its graph 4 + 1/x+1

1. Find x-intercepts by converting the function into 1 function 4(x+1)/ x+1 + 1/(x+1) = 4(x+1) + 1 / (x+1) 4x + 4 + 1 / (x+1) 4x + 5 / x + 1 ; x-int: x = -5/4 2. Find y intercept Plug 0 in for x y-int = 5 3. Find VA VA: x = -1 lim x --> -1 - f(x) = -inf lim x --> -1 + f(x) = inf 4. Find HA (Compare degree of leading terms) num = denom --> 0 / 1 = 0 + 4 --> 4 HA: y =4 Refer to graph on Asymptotes and Intercepts #9

Using only algebra, find the cubic function with the given table of values. x = -4 , 0 , 3 , 4 y = 0, 96, 0, 0

1. Find zeros where y = 0 (x + 4) (x - 3) (x-4) 2. Multiply and simplify x^3 - 3x^2 - 16x + 48 3. Using table we know yint is 96, satisfy condition by multiplying by 2 2x^3 - 6x^2 - 32x + 96

Solve the inequality 1/x+4 + 1/x-8 <= 0

1. Identify LCD (x-8)(x+4) 2. Multiply each term by LCD 3. Zeros x = 2 4. Undefined x = 8, x = -4 5. Sign Chart 6. f(x) <= 0: (-inf, -4)U[2,8)

Solve algebraically 2 - (5/x-4) = (20/x^2 +4x)

1. Identify LCD --> (x)(x+4) 2. Multiply each term by LCD 2x(x+4) - x = 4 3. Get all terms on 1 side 2x^2 + 7x - 4 =0 4. Solve for the zeros (2x-1)(x+4) x = 1/2, x = -4 5. Check for extraneous solutions x = -4 is an extraneous solution x = 1/2 is a solution {1/2}

Solve the equation algebraically (7x / x+5) + (1/x-2) = (7 / x^2+3x-10)

1. Identify LCD --> (x+5)(x-2) 2. Multiply each term by LCD 7x(x-2) + x+5 = 7 3. Get all terms on 1 side 7x^2 - 13x - 2 =0 4. Solve for the zeros (7x+1)(x-2) x = -1/7, x = 2 5. Check for extraneous solutions x = 2 is an extraneous solution x = -1/7 is a solution {-1/7}

i^2 =

-1

Base points of an exponential function

(0,1) and (1,b)

Base points for logarithmic functions

(1,0) and (b,1)

Determine the x value that causes the polynomial to a) be zero b) be positive c) be negative f(x) = (5x^2 + 9)(x-6)^2(x+2)^3

*Note: b/c 5x^2 +9 is squared inside the parentheses and squares can only result in non-negative numbers, there is no zero for this factor a) Zeros: x = 6, x = -2 b) f(x) > 0: (-2, 6) U (6, inf) c) f(x) < 0: (-inf, -2) 0 0 <------------|------------------------|-----------------> -2 6 Test conditions x = -3 (+)(+)(-) == - x = 0 (+)(+)(+) == + x = 7 (+)(+)(+) == +

Compute the exact value of the function for the given x-value f(x) = -5*2^x for x = 1/3

-5*cbrt(2)

Solve the equation and check your answer q + 4 q-5 1 ------- + ------ = ----- 2 2 2

1. Identify LCD --> 2 2. Multiply each term by LCD q+4 + q-5 = 1 3. Get all terms on 1 side 2q-2 =0 4. Solve for the zeros q = 1 5. Check for extraneous solutions q = 1 is a solution {1}

Solve the equation and check your answer q + 10 q-11 1 ------- + ------ = ----- 6 6 6

1. Identify LCD --> 6 2. Multiply each term by LCD q+10 + q-11 = 1 3. Get all terms on 1 side 2q-2 =0 4. Solve for the zeros q = 1 5. Check for extraneous solutions q = 1 is a solution {1}

Solve the equation algebraically r+3 = 28/r

1. Identify LCD --> r 2. Multiply each term by LCD r^2 + 3r = 28 3. Get all terms on 1 side r^2 + 3r -28 =0 4. Solve for the zeros (r +7) (r -4) r = -7, r= 4 5. Check for extraneous solutions Both r=-7, and r=4 are a solutions {4,-7}

Solve (2/x-1) + x = 11

1. Identify LCD --> x-1 2. Multiply each term by LCD 2 + x(x-1) = 11(x-1) 3. Get all terms on 1 side x^2 - 12x + 13 =0 4. Solve for the zeros x = -(-12) +- sqrt((-12)^2 - (4 * 1 * 13) ---------------------------------- 2(1) x = 12 +- sqrt(92) ------------- 2 x= 6 +- sqr(23) 5. Check for extraneous solutions Both are solutions {6+sqrt(23), 6-sqrt(23)}

Solve: (3/x-1) + x = 9

1. Identify LCD --> x-1 2. Multiply each term by LCD 3 + x(x-1) = 9(x-1) 3. Get all terms on 1 side x^2 - 10x + 12 =0 4. Solve for the zeros x = -(-10) +- sqrt((-10)^2 - (4 * 1 * 12) ---------------------------------- 2(1) x = 10 +- sqrt(52) ------------- 2 x= 5 +- sqr(13) 5. Check for extraneous solutions Both are Solutions {5+ sqrt(13), 5-sqrt(13)}

Solve the equation algebraically x + 9x/x-8 = 72/x-8

1. Identify LCD --> x-8 2. Multiply each term by LCD x^2 -8x + 9x = 72 3. Get all terms on 1 side x^2 + x -72 =0 4. Solve for the zeros (x+9)(x-8) x = -9, x = 8 5. Check for extraneous solutions x = 8 is an extraneous solution x = -9 is a solution {-9}

Solve: (2y/y+2) + (4/y) + 2 = (8/y^2 + 2y)

1. Identify LCD --> y(y+2) 2. Multiply each term by LCD 2y(y) + 4(y+2) = 8 3. Get all terms on 1 side 4y^2 + 8y =0 4. Solve for the zeros (4y) (y+2) y = 0, y = -2 5. Check for extraneous solutions Both y = 0, and y = -2 are extraneous solutions No Solution

Graph the function below f(x) = -ln(x-5)

1. Identify Transformations Right 5 units, reflect across the x-axis 2. Identify VA x-5 = 0 x = 5 3. Apply to Base points of logarithms (1,0) --> (6,0) (e, 1) --> (e+5 , -1) 4. Graph

Graph the polynomial function. f(x) = 44x^4 - 61x^3-6x^2 + 19x + 4

1. Identify basic shape based on degree 2. Identify end behavior 3. Find y intercept : 4 Graph

Find the remainder when f(x) = 8x^26 - 3x is divided by x+1

1. Identify k --> -1 2. Use the remainder therom to find the remainder by plugging k in for x 3. remainder is 11

Express as a difference of logarithms log 66/y

1. Identify rule log(R/S) = log R - log S 2. log 66 - log y

To solve equations in one variable

1. Identify the LCD 2. Multiply each term by LCD 3. Get all terms on 1 side 4. Solve for the zeros 5. Check for extraneous solutions by plugging zero in for the x in the LCD

Use the properties of logarithms to write an expanded expression log<5>y^2

1. Identify the rule log<b> R^c = c * log<b> R 2. 2 * log<5> y

Describe how to transform the graph f(x) = x^n into the graph of the given polynomial function: g(x) = -1/2(x+4)^3 + 1; Sketch the transformed graph; Find the y intercept

1. Identify transformations and place them from inside out Left 4, Vertical Shrink of magnitude 1/2, Reflection over the x axis, up 1 2. Find y intercept by plugging 0 in for x -1/2(0+4)^3+1 -1/2(4)^3 +1 -1/2 (64) + 1 -32 + 1 y = -31 yint = (0, -31) 3. Refer to Assignment Polynomials of Higher Degrees for Sketch

Describe how to transform the graph f(x) = x^n into the graph of the given polynomial function: g(x) = 2(x+5)^3; Sketch the transformed graph

1. Identify transformations and place them from inside out Left 5, Vertical Stretch of magnitude 2 2. Refer to Assignment Polynomials of Higher Degrees for Sketch

Use the properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms ln(25d)

1. Identify which propert log(R*S) = log R + log S 2. Split ln 25 + ln d 3. Reduce ln 5^2 + ln d 2 ln 5 + ln d

Graph (calculator) the function in a viewing window that shows all of its extrema and x intercepts. Describe the end behavior using limits. zeros: (x-2), (x+3), (x+5)

1. Identify x int 2, -3, -5 2. Factor and plug into calculator 3. Draw picture with viewing window 4. Describe using end behavior lim x --> inf f(x) = inf lim x --> -inf f(x) = -inf

Find the exact solution algebriacally log<6>(2x-7) = 4

1. If solving a logarithm with a argument = to something 6^4 = 2x-7 1296 = 2x-7 2x = 1303 x = 651.5 {651.5}

1.23^x = 5.7

1. If solving and exponent is a variable log 1.23^x = log 5.7 x * log 1.23 = log 5.7 x = log 5.7 / log 1.23 x = 8.408 {8.408}

Describe how to transform the graph of f(x) into the graph of g(x) Sketch the graph by hand and support answer with a grapher f(X) = e^x g(x) = e^-4x

1. List out transformations Horizonatly shrink by a factor of 1/4, Reflect over the y-axis 2. Apply transformations to the base points of an exponential function (0,1) and (1,b) (0,1) --> (0, 1) (1,b) --> (1,e) --> (-1/4 , -e) 3.Graph

Describe how to transform the graph of f(x) into the graph of g(x) Sketch the graph by hand and support answer with a grapher f(X) = 3^x g(x) = 3^x-3

1. List out transformations Right 3 units 2. Apply transformations to the base points of an exponential function (0,1) and (1,b) (0,1) --> (3,1) (1,b) --> (1,3) --> (4,3) 3. Graph

Solve for x e^x - e^-x ------------ = 5 2

1. Multiply by 2 e^x - e^-x = 10 2. Multiply each term by [e]^x [e]^2x - 1 = 10[e]^x 3. Rearrange to a Quadratic [e]^2x - 10[e]^x - 1 4. Quadtratic formula [e]^x = Y Y^2 - 10Y - 1 Y= 5 +- sqrt(26) [e]^x = 5 + sqr(26) [e]^x = 5 - sqrt(26) ---> x can't be negative e^x = 5 + sqrt(26) ln 5 + sqrt(26) = x x = 2.312

Rules for Synthetic and Long division

1. Must be written in standard form 2. All terms must be present

State whether the function is exponential growth or decay and describe using limits f(x) = 1/3^-x

1. Must covert into a*b^x 1/3^-x 3^x 2. Compare to rules Growth: a>0 b>1 Decay: a>0 0<b<1 Exponential Growth lim x--> -inf f(x) = inf lim x--> inf f(x) = 0

Describe the end behavior of the polynomial function using end behavior notation: f(x) = 6x^2 - x^3 + 4x - 2

1. Rearange -x^3 + 6x^2 + 4x -2 2. Describe using end behavior LC = -1 ; Negative = reflection, so opposite than normal Degree = 3; Odd degree = different directions lim x --> -inf f(x) = inf lim x --> inf f(x) = -inf

Graph the function below f(x) = 4 + log(x)

1. Rewrite log(x) + 4 2. Identify transformations Up 4 units 3. Identify VA x = 0 3. Apply transformation to base points (1, 0) --> (1, 4) (10, 1) --> (10, 8) 4. Graph

Graph the function below f(x) = log(-3 - x)

1. Rewrite -(3 + x) --> -(x + 3) --> log(-x - 3) 2.Identify transformations Right 3, Reflect across y-axis 2.5 Identify VA -x-3 = 0 -x = 3 x = -3 3. Apply to base points of logartihms (1,0) and (b,1) (1,0) --> (-4,1) (10,1) --> (-13, 1) 4. Graph

Find the zeros of the function algebraically: f(x) = 7x^3 - 26x^2 - 8x

1. See if you can factor out anything x (7x^2 - 26x - 8) 2. Factor x (7x + 2) (x -4) 3. Set factors equal to 0 x = 0 7x + 2 = 0 x = -2/7 x-4 = 0 x = 4

Find the zeros of the function algebraically: f(x) = x^2 + x - 30

1. See if you can factor out anything - no 1.5 Factor (x + 6) (x - 5 ) 2. Set factors equal to zeros x + 6 = 0 x = -6 x - 5 = 0 x = 5

Find the zeros of the function algebraically: f(x) = 6x^2 + 25x - 9

1. See if you can factor out anything - no 2. Factor (3x - 1) (2x + 9) 3. Set factors equal to zero 3x - 1 = 0 x = 1/3 2x+9 = 0 x = -9/2

Solve ln(3x-2) + ln(x-1) = 2 ln x

1. Simplify ln (3x-2 * x-1) = ln x^2 2. Same base logarithms = each other (3x-2 * x-1) = x^2 3x^2 - 5x + 2 = x^2 2x^2 - 5x + 2 = 0 (2x - 1) (x -2) x = 2, x = 1/2 (extraneous) {2}

State the degree and list the zeros of the polynomial function. State the multiplicity of each zero, and whether the graph crosses the x-axis at the location Sketch a graph f(x) = x (x-3)^2

1. State the degree by adding the multiplicities of each factor 2 + 1 = 3 degree = 3 2. State the multiplicity of each zero and whether it crosses the x-axis Multilicity of x = 1; crosses x axis because it is odd Multiplicity of (x-3)^2 = 2; touches x axis because even 3. Reference sheet for graph

How to find end behavior asymptotes

1. Sythetically divide the numerator by the denominator and find the remainder 2. Drop the remainder and the rest of the division is the end behavior asymptotes

Write the expression using only natural logarithms log<5> x

1. Use change of base ln x ---- ln 5

Write the expression using only common logs log<5> x

1. Use change of base log x ----- log 5

Describe how to transform the graph of ln x into the function given below f(x) = log<1/4> x

1. Use change of base to see how ln x turned into log<1/4> x ln (x) / ln (1/4) ln(x) / ln 1 - ln 1/4 --> ln(x) / -ln(4) -1/ln(4) * (ln x) 2. Identify Transformations Vertical shrink by a factor of 1/ln(4), and reflected across the x-axis 3. Identify VA: x = 0 4. Apply Transformations (1,0) --> (1 , 0) (e, 1) --> (e , -0.721) 5. Graph

How to Find Undefined Values

1. What makes the number underneath the radical negative If it makes a radical negative you use an in equality such as x < # to describe it, then proceed to cross out all the numbers behind it. 2. What makes the denominator zero

Evaluate the logarithimic expression: log 1/10

1. Write as exponential 1/10 = 10^x 10^-1 = 1/10 log 1/10 = -1

Using only algebra, find a cubic function with the given zeros: -1 , 2 , -6

1. Write as factors (x + 1) ( x - 2) ( x + 6) 2. Multiply x^3 + 5x^2 -8x - 12

Using only algebra, find a cubic function with the given zeros: sqrt(3) , -sqrt(3), 5

1. Write as factors (x - sqrt(3) ) (x + sqrt(3) ) (x - 5) 2. Multiply x^3 - 5x^2 - 3x + 15

Describe how the graph of f(x) = 3x-4/x + 2 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function

1. Write as vertex form synthetically divide the numerator by denominator 3 + -10/x+ 2 --> -10(1/x+2) + 3 From 1/x left 2 units, followed by a vertical stretch by a factor of 10, then a reflection across the x-axis, and then up 3 units 2. Identify asymptotes and their limits VA: x = -2 HA: y = 3 lim x --> -2 - f(x) = inf lim x --> -2 + f(x) = -inf lim x --> -inf f(x) = 3 lim x --> inf f(x) = 3 3. Sketch Refer to graphs of rational functions #7

Describe how the graph of f(x) = 3x-1/x+1 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function

1. Write as vertex form synthetically divide the numerator by denominator 3 - 4/x+ 1 --> -4(1/x+1) + 3 From 1/x left 1 units, followed by a vertical stretch by a factor of 4, then a reflection across the x-axis, and then up 3 units 2. Identify asymptotes and their limits VA: x = -1 HA: y = 3 lim x --> -1 - f(x) = inf lim x --> -1 + f(x) = -inf lim x --> -inf f(x) = 3 lim x --> inf f(x) = 3 3. Sketch Refer to graphs of rational functions #8

Determine the formula for the exponential whose values are given (0,5/4) and (1,5/16)

1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 5/4 = a *b^0 (b^0 is 1) 5/4 = a 3. Solve for b using the other point and a plugged in 5/16 = 5/4 * b^1 1/4 = b^1 b = 1/4 4.Write as the function f(x) = 5/4 * 1/4^x

Determine the formula for the exponential whose values are given (0,7) and (2,14)

1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 7 = a *b^0 (b^0 is 1) a = 7 3. Solve for b using the other point and a plugged in 14 = 7 * b^2 2 = b^2 b = sqrt(2) 4.Write as the function f(x) = 7 * 2^x/2

Determine a formula for the exponential function with points (0, 2.5) and (1,4.7)

1. Write formula: P(t) = P(1 + r)^t 2. Plug in (0, 2.5) 2.5 = P(1+r)^0 P = 2.5 3. Plug in (1, 4.7) 4.7 = 2.5(1+r)^1 r = .88 4. Write as function P(t) = 2.5(1.88)^t

Write a polynomial function of minimum degree in standard form with real coefficents whose zeros and their multiplicities include those listed: 1 (multiplicity of 2) -5 (multiplicity of 3)

1. Write out all factors (x-1)(x-1)(x+5)(x+5)(x+5) 2. Multiply and Simplify x^5 + 13x^4 + 46x^3 -10x^2 -175x +125

For the following, find function f(x) defined by a polynomial of degree 3 with real coefficents that satisfy the given conditions: Zeros: -3, -1, 4 LC: 6

1. Write zeros as factors (x +3) (x + 1) ( x- $) 2. Multiply and simplify x^3 - 13x -12 3. Apply conditions 6(x^3 - 13x -12) 6x^3 -78x - 72

Use the Rational Zero Therom to write a list of all potential rational zeros. Then determine which ones if any are real zeros f(x) = x^3 - 6x^2 - 9x + 14

1. p/q --> 14/1 1 2 7 14 -------- 1 possible rational zeros +/- 1, 2, 7, 14 actual zeros are 1, 7, -2

Evaluate the logarithimic expression: log<6> 1296

1. write as exponential 1296 = 6^x 6^4 = 1296 log<6> 1296 = 4

Factor the polynomial and solve the inequality using a sign chart f(x) = 4x^3 - 7x^2 -21x + 18 >= 0

1.Synthetically divide to find the zeros x = 3, x = -2, x = 3/4 2. Plot on sign chart and test given conditions x = 3 (-)(+) == - x = 0 (-)(-) == + x = 1 (-)(+) == - x = 4 (+)(+) == + 3. f(x) >= 0: [-2, 3/4] U [3, inf)

The recipricol function

1/x

Compute the exact value of the function for the given x-value f(x) = 5*2^x for x = 0

5

long division of polynomials

1polynomial / 2polynomial 1p first term / 2p first term = thing on top multiply thing on top by 2p subtract result from original polynomial repeat with left over polynomial

Use synthetic division to perform each division 2x^3 + 10x^2 + 8c - 17 ----------------------- c + 3

2x^2 + 4x - 4 - 5/c+3 -5/c+3 is the remainder

Find the unique polynomial with real coefficents that meets these conditions: Degree = 4 Zeros: 1, -4, 2 - i; f(0) = -60

3x^4 - 3x^3 -33x^2 + 93x -60

Basic Domian signs with solving inequalities

<= or >= will usually have a [ and a ) < or > will have ( and )

End Behavior: Even Degree~_______ direction(s); Odd Degree~_______ direction(s)

Even = Same Odd = Opposite

Irredusible Quadratic

A quadratic that results in a imaginary number Use sqrt( b^2 -4ac) to determine if it is a irredusible quadratic

An upper bound is represented by

All Non-negative numbers

A lower bound is represented by

Alternating non-negatives and non-positives

Odd multiplicty does what?

Crosses the x axis

Find the domain of the function f(x). Use limits to describe the end behavior at values of x not in the domain. f(x) = 1/x+7

D: (-inf, -7) U (-7, inf) lim x --> -7- f(x) = -inf lim x --> -7+ f(x) = inf

Find the domain of the function f(x). Use limits to describe the end behavior at values of x not in the domain. f(x) = -1/x^2 -16

D:(-inf, -4) U (-4,4) U (4, inf) lim x --> -4 - f(x) = -inf lim x --> -4 + f(x) = inf lim x --> 4 - f(x) = inf lim x --> 4+ f(x) = -inf

How to find horizontal asymptotes and their format

Find by comparing the degrees of the numerator and denominator If numerator < denominator then y = 0 is a HA If numerator = denominator then y = LC num / LC denom If numerator > denominator then No Horizontal Asymptote, but need to look for End Behavior Asymptote HA: y = #

How to find vertical asymptotes and their format

Find by setting the denominator = to 0 Once found make sure that the numerator is not 0 when plugging it in for x. If it does then it is not a VA VA: x = #

How to find x intercepts of rational functions and their format

Find x-int by setting the numerator = to 0 x-ints: x = #, x = #

How to find y intercepts of rational functions and their format

Find y-int by plugging 0 in for x y int: y = #

Describe how the graph of f(x) = 2/x-3 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function

From 1/x right 3 units, then vertically stretch by a factor of 2. VA: x = 3 HA: y = 0 lim x --> 3 - f(x) = -inf lim x --> 3 + f(x) = inf lim x --> inf f(x) = 0 lim x --> -inf f(x) = 0 Refer to graphs of rational functions for sketch #5

Describe how the graph of f(x) = 2/x-6 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function

From 1/x right 6 units followed by a vertical stretch by a factor of 2. VA: x = 6 HA: y = 0 lim x --> 6 - f(x) = -inf lim x --> 6+ f(x) = inf lim x --> inf f(x) = 0 lim x --> -inf f(x) = 0 Refer to graphs of rational functions for sketch #6

Horizontal asymptote format

HA: y = #

In an exponential function there is 1 _____ but both ends ________ it

Horizontal Asymptote don't

The remainder therom

If f(x) / x - k then f(k) will give you the remainder

Is the following function exponential? If so, state the initial value and base. If not explain why y = 5^z

Is an exponential function inital value is 1 (a) base is 5

Use synthetic division to check that the number k is an lower bound for the real zeros of the function: f(x) = 5x^3 - 7x^2 + x -5 k = -7

It is a lower bound Alternating non-positive numbers

Use synthetic division to check that the number k is an upper bound for the real zeros of the function: F(x) = 6x^3 - 7x^2 + x - 6 k = 3

It is an upper bound all non-negative numbers

What is multiplicity?

Multiplicity is the exponent of a factor

If all zeros are extraneous solutions the function is said to have ...

No Solution

Is the following function exponential? If so, state the initial value and base. If not explain why y = z^9

Not an exponential function because the base is a variable

Determine the exponential function that satisfies the given conditions Initial Value: 0.6 halving every 5 days

P(t) = .6(.5)^t/5 If being halfed every ___ # of days divide the t variable by it

Determine the exponential function that satisfies the given conditions Initial Value: 16 decreasing at a rate of 60% per year

P(t) = 16(1-.60)^t P(t) = 16(.4)^t

Determine the exponential function that satisfies the given conditions Initial Value: 5 increasing at a rate of 18% per year

P(t) = 5(1+.18)^t P(t)=5(1.18)^t

Exponential Modeling and conditions

P(t) = P(1+r)^t r>0 exponential growth r<0 exponential decay r must be a decimal

Use the remainder therom to find the remainder when f(x) is divided by k. f(x) = x^3 - 3x^2 + 4 k = -2

Remainder = -16

Find rational zeros by using

The Rational Zero Therom gives you the possible rational zeros of a polynomial

Even multiplicity does what?

Touches the x axis

If log<b>U = log<b>V then

U = V

Evaluate the logarithimic expression: ln (e^-5)

Use ln e^n = n ln (e^-5) = -5

Change of Base -Uses and formula

Used to change logs with a base to common or natural logs log<b> N = (log<c> N) / (log<c> b)

Evaluate the logarithimic expression: log<11> 11

Using log<b> b = 1 log<11> 11 = 1

Find the horizontal and vertical asymptotes of f(x). Use limits to describe the cooresponding behavior. f(x) = 5x^2 + 2 / x^2 + 4

VA: None HA: y = 5 lim x --> -inf f(x) = 5 lim x --> inf f(x) = 5

Vertical asymptote format

VA: x = #

Find all real zeros of the function, finding exact values whenever possible, identify each zero as rational or irrational: f(x) = 7x^3 - 2x^2-35x + 10

Zeros: 2/7 rational +/- sqrt(5) irrational

Solve log x^2 = 2

When dealing with solving, don't use exponential logarithm properties 10^2 = x^2 100 = x^2 x = +- 10

For the following polynomial, one zero is given. Find the remaining zeros: f(x) = x^4 + 7x^2 - 144 Zero: 4i

Zeros: 4i, -4i, 3, -3

If a > 0 and b > 1 the function is an ________________ function add limits

exponential growth lim x-->-inf f(x) = 0 lim x --> inf f(x) = inf

Use the factor therom to determine whether the first polynomial is a factor of the second polynomial x + 6 ; x^3 + 7x^2 + 4x - 12

Yes it is a factor

Write the function as a product of linear and irredusible quadratic factors with all real coefficents f(x) = x^3 -14x^2 -8x -105

Zero: 15 Factor form: (x - 15)(x^2 + x + 7)

Find all of the zeros and write a linear factorization of the function: f(x) = x^3 - 11x^2 + 9x -99

Zeros: 11, +- 3i Factorization: (x-11)(x-3i)(x+3i)

If the function has an end behavior with a number then it implies that the function has ...

a horizontal asymptote

An extraneous solution is

a zero that makes the LCD = 0

Factored form format

a(x-k)(x-k)(x-k)

Determine the x value that causes the polynomial to a) be zero b) be positive c) be negative f(x) = (x+4)(x+2)(x-1)

a) Zeros: x = -4, x = -2, x = 1 b) f(x) > 0: (-4, -2)U(1, inf) c) f(x) < 0: (-inf, -4) U (-2, 1) 0 0 0 <------|---------|-----------------|--------------------> -4 -2 1 Test conditions x = -5 (-)(-)(-) == - x = -3 (+)(-)(-) == + x = 0 (+)(+)(-) == - x = 2 (+)(+)(+) == +

Domain lists _____________

all possible x values

Range lists __________________

all possible y values

Basic Form of a Polynomial Function and rules

ax^n + bx^n-1 + cx^n-2 ... + yx + z - a can never be 0 -coefficents must be real numbers

1/b^# is the same as

b^-#

Evaluate the logarithimic expression: 6^log<6> 2

b^log<b> m = m 6^log<6> 2 = 2

Vertical asymptote end behavior rules

choose a number close to the asymptote on the left, and plug it in to the numerator to determine if it is positive or negative. Do the same to the denominator. Determine if the entire function is positive or negative with the number if positive = positive inf if negative = negative inf Repeat for right side

Complex zeros are always found in

conjugate pairs 2 irrational 1 rational or 2 irrattional 2 rational etc.

If a >0 and 0<b<1 the function is an ______________ function add limits

exponential decay lim x-->-inf f(x) = inf lim x--> inf f(x) = 0

Exponetial function formula and conditions

f(x) = a * b^x a cannot = 0 b > 1

The factor therom

f(x) will have a factor of x-k if f(k) = 0

The number of complex zeros a function has is based on the

functions degree

lim x --> #+ f(x) = -inf

implies that as x approaches a number from teh right side it goes towards negative infinity

lim x --> #- f(x) = inf

implies that as x approaches a number from the left side it goes towards infinity

If a function has >= or <= and two zeros in the same direction in a row you use

inf and -inf

Format of writing end behavior

lim x --> -inf / inf f(x) = inf / -inf

Horizontal aysmptote end behavior rules and format

lim x --> inf f(x) = # lim x --> -inf f(x) = # Rules plug in a large value for x and determine if it is a horizontal asymptote by comparing it with the horizontal asymptote test

Evaluate the logarithimic expressionln ln 9throot(e^7)

ln (e^7)^1/9 ln (e^7/9) ln 9throot(e^7) = 7/9

Use the properties of logarithms to expand the logarithmic expression as much as possible log cbrt(x/z)

log (x/z) ^ 1/3 log (x^1/3 / z^1/3) log x^1/3 - log z^1/3 1/3 log x - 1/3 log z

Assuming x, y, z are positive use the properties of logarithms to write the expression as a single logarithm 8 log (xy) - 7 log(yz)

log(xy)^8 - log(yz)^7 log(x^8 y^8) - log(y^7 z^7) log(x^8 y^8) / (y^7 z^7) log(x^8 y) / z^7

Properties of all logarithmic functions

log<b> (R*S) = log<b>R + log<b>S log<b>(R/S) = log<b>R - log<b>S log<b> R^c = c * log<b> R

Properties of Log, Natural Logs, Common Logs

log<b> 1 = 0 ; ln 1 = 0; log 1 = 0 log<b> b = 1; ln e = 1; log 10 = 1 log<b> b^n = n; ln e^n = n; log 10^n = n b^log<b> m = m; e^ ln m = m; 10 ^ log m = m

When writing out a linear factorization with just factors ...

make sure to include an a value that causes the y-intercepts to be comparable

Rational zeros Therom

narrows down the possible rational zeros p/q p is the constant q is the leading coefficent

logarithm arguments cannot be

negative

as numbers approach a vertical asymptote from either side and go down the end behavior is ...

negative infinity

Divide f(x) by d(x) and write a summary statement in polynomial form and fraction form f(x) = x^2 -14x + 57 d(x) = x-7

polynomial form: x^2 -14x + 57 = (x-7)(x-7) + 8

as numbers approach a vertical asymptote from either side and go up the end behavior is ...

positive infinity

Fraction form: Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1

problem = quotient + remainder /divisor Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 2x^4-x^3 - 2 ------------- = x^2 - x + ( x-2 / 2x^2 + x + 1) 2x^2 + x + 1

i =

sqrt(-1)

Multiplying two of the same radical =

the number underneath the radical

polynomial form: Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1

top = divisor * quotient + remainder Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 2x^4 - x^3 -2 = (2x^2 + x +1) * (x^2 - x) + (x -2)

If b^u = b^v then

u = v

f(x) > 0 definition

when the function has y values greater than 0

f(x) >= 0

when the function has y values greater than or equal to 0

f(x) < 0

when the function has y values less than 0

f(x) <= 0

when the function has y values less than or equal to 0

Solve for x log x = 2

x = 10^2 x = 100

when given an exponent of 1/x it is the same as

x root of base

Find the intercepts, vertical asymptotes, and the end behavior asymptote and graph the function together with its end behavior asymptotes f(x) = x^4 + 3 / x + 1

x-int: None y-int: y = 3 VA: x = -1 lim x --> -1 - f(x) = -inf lim x --> -1 + f(x) = inf HA: None EBA: x^3 - x^2 + x -1 Refer to graph on Asymptotes and Intercepts #14

Graph the function f(x) = x+8 / x^2 -4x -32 Find all asymptotes. List x and y intercepts

x-int: x = -8 y-int: y = -1/4 VA: x = -4, x= 8 lim x --> -4 - f(x) = inf lim x --> -4 + f(x) = -inf lim x --> 8 - f(x) = -inf lim x --> 8 + f(x) = inf HA: y = 0 lim x --> -inf f(x) = 0 lim x --> inf f(x) = 0 Refer to graph on Asymptotes and Intercepts # 3

Graph the function f(x) = 11x^2 + x -11 / x^2 -1 Find all asymptotes. List x and y intercepts

x-int: x = 0.96, x = -1.05 y-int: y = 11 VA: x = 1, x = -1 lim x --> 1 - f(x) = -inf lim x --> 1 + f(x) = inf lim x --> -1 - f(x) = -inf lim x --> -1 + f(x) = inf HA: y = 11 lim x --> -inf f(x) = 11 lim x --> inf f(x) = 11 Refer to graph on Asymptotes and Intercepts # 5

Write the polynomial in standard form, and identify the zeros of the function and the x-ints of the graph f(x) = (x - 9i)(x + 9i)

x^2 + 81 --> standard form zeros: +/- 9i (no x intercepts because imiginary numbers)

Key to logartithms

y = a *b^x x = log<b> y

synthetic division of polynomials

zero of divisor | term1 term 2 term 3 1. Bring down 1 2. Multiply by zero 3. Add to term 2 4. Repeat 5. Last number is the remainder

Irrational zeros are

zeros not on the list