GAVS Algebra II
Write the equation of the piece-wise function graphed below:
f(x)={3,−3≤x<0 {x,0≤x≤3
A polynomial function with a degree of 3 is called a Quartic Function.
False
The domain of the graph of this equation is (−1.1026,∞). y=x^4-4x^3+\frac{14}{3}x^2-\frac{9}{7}x+2y=x4−4x3+314x2−79x+2
False
The range of the graph of this equation is all real numbers. y=-4x^4-3x^2-19
False
Jimmie invested $13,000 at 5.23% compounded monthly. What will Mike's account balance be in 42 years?
$116,370.20
Mike invested $29,000 at 7.24% compounded daily. What will Mike's account balance be in 21 years?
$132,627.58
The graph below represents the cost of a cellphone given the number of minutes used. What is the cost of the phone if you use 700 minutes?
$80
(−𝟏𝟐𝟏 + 𝟒𝟑𝒊) + (𝟗𝟐 − 𝟗𝟗𝒊)
-29-56i
A grocery store chain has been tracking data on the number of shoppers that use coupons. The data shows that 71% of all shoppers use coupons. 36 times out of 40 these results were considered accurate to within 2.5%. What is the confidence interval?
.685 to .735
An equation for loudness, in decibels, is L\;=\;10\;log_{10}\;RL=10log10R, where R is the relative intensity of the sound. Sounds that reach levels of 120 decibels or more are painful to humans. What is the relative intensity of 120 decibels?
10^12
Simplify √121
11
(1-3i)(2+5i)
17-i
(𝟖 + √−𝟖𝟏) − (−𝟏𝟎 + √−𝟏𝟎𝟎)
18-i
The graph below shows the number of employees needed at a daycare based on the number of children present. How many employees are needed if there are 18 children?
4 employees
Given the function of y=x(x-2)^3(x+1)^2(x-1)^2 what would be the maximum number changes of direction (turning points)?
7
Determine if the following function is even, odd or neither. f(x)=3x^8+6x^4+1
Even
(0, 0.75) and (-0.634, 0) are the x- and y-intercepts of the following polynomial equation: y=-2x^4-2x^2-3x+6x^3+\frac3/4
False
What are the Intervals of Increase and Decrease in the following Graph? f1(x)=-3*2^(x-5)-4
Intervals of Increase: none Intervals of Decrease: (-infinity, infinity)
Bias - A mistake causing results that are not representative of the population. Center - Measures of center refer to the summary measures used to describe the most "typical" value in a set of data. The two most common measures of center are median and the mean. Census - A census occurs when everyone in the population is contacted. Central Limit Theorem - The CLT allows us to use normal calculations to determine probabilities about sample proportions and sample means obtained from populations that are not normally distributed. Confidence Interval - An interval for a parameter, calculated from the data, usually in the form estimate ± margin of error. The confidence level gives the probability that the interval will capture the true parameter value in repeated samples. Continuous Random Variables - Have an infinite number of possible values. Data - Information about a product or process, usually in numerical form. Descriptive Statistics —This involves the organization, summarization, and display of data. Discrete Random Variables - Have a finite number of distinct values or counts. Empirical Rule - If a distribution is normal, then approximately: 68% of the data will be located within one standard deviation symmetric to the mean. 95% of the data will be located within two standard deviations symmetric to the mean. 99.7% of the data will be located within three standard deviations symmetric to the mean. Frequency Distribution - Instead of listing every data point, a frequency distribution will list the value with its associated frequency. Inferential Statistics —This involves using a sample to draw conclusions about a population. Interquartile Range - The difference between the first and third quartile values. This IQR measures the spread of the middle half of the data. Margin of Error - The value in the confidence interval that says how accurate we believe our estimate of the parameter to be. The margin of error is comprised of the product of the z-score and the standard deviation (or standard error of the estimate). The margin of error can be decreased by increasing the sample size or decreasing the confidence level. Mean Absolute Deviation - A measure of variation in a set of numerical data, computed by adding the distances between each data value and the mean, then dividing by the number of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation is 20. Normal Distribution - A frequency distribution that is the symmetric, bell-shaped curve and has the data spread evenly around a specific center. Outliers - Data that are far away from most of the data points. Parameters - These are numerical values that describe the population. The population mean is symbolically represented by the parameter . The population standard deviation is symbolically represented by the parameter . Population - The entire set of items from which data can be selected. Qualitative Data - Consist of attributes, labels, or non-numerical entries. Quantitative Data - Consist of numerical measurements or count. Quartiles - Divide an ordered data set into four equal parts. Random - Events are random when individual outcomes are uncertain. However, there is a regular distribution of outcomes in a large number of repetitions. Range - The difference between the greatest data element and the least data element. Sample - A subset, or portion, of the population. Sample Mean - A statistic measuring the average of the observations in the sample. It is written as . The mean of the population, a parameter, is written as . Sample Proportion - A statistic indicating the proportion of successes in a particular sample. It is written as . The population proportion, a parameter, is written as p. Sampling Distribution - A statistics is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Sampling Variability - The fact that the value of a statistic varies in repeated random sampling. Shape -The shape of a distribution is described by symmetry, number of peaks, direction of skew, or uniformity. Spread - The spread of a distribution refers to the variability of the data. If the data cluster around a single central value, the spread is smaller. The further the observations fall from the center, the greater the spread or variability of the set. (range, interquartile range, Mean Absolute Deviation, and Standard Deviation measure the spread of data). Standard Deviation - The square root of the variance. Statistic - A numerical description of a sample characteristic. Statistics - The science of collecting, organizing, and interpreting data in order to make decisions. Survey - An investigation on one or more characteristic of a population, either through census or sampling. Variance - The average of the squares of the deviations of the observations from their mean.
Inferences and Conclusions from Data Key Words
Determine if the following function is even, odd or neither. g(x)=6x^5+2x^3-16x
Odd
The value of a car after 5 years is what type of data?
Quantitative
The transformation y=f(x) + k does what to the graph?
Raise graph by k units
What is the shape of this histogram?
Skewed Left
A polynomial function with a degree of 0 is called a Constant Function.
True
A rough sketch of y=x^3-4x^2+3xy=x3−4x2+3x is "S" shaped.
True
The process used to collect data will __________ of the study.
affect the validity and the bias
Normal distribution models what type of variable?
random continuous variable
Mark is investing $47,000 in an account paying 5.26% interest compounded continuously. What will Mark's account balance be in 17 years?
$114,932.80
(2-3i)-(6-18i)
-4+15i
(−𝟏𝟏 + √−𝟒) + (𝟒 + √−𝟑𝟔)
-7+8i
An earthquake rated at 3.5 on the Richter scale is felt by many people, and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude reading m is given by m=log_{10}\;xm=log10x, where x represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures 4.5 on the Richter scale than one that measures 3.5?
10 times
What is the equation of the following graph?
2 - √(x+3)
b^2 - 4ac > 0 b^2 - 4ac = 0 b^2 - 4ac < 0
2 Real #s 1 Real # 2 Imaginary #s
A report sponsored by the California Citrus Commission concluded that cholesterol levels can be lowered by drinking at least one glass of a citrus product each day. This report is biased. The report may be biased because:
A report sponsored by the citrus industry is much more likely to reach favorable conclusions.
The number of hours that college students sleep on a week night is approximated by a normal curve with a mean of 7 hours and a standard deviation of 1.7 hours. Answer the question below using a z table: On a week night, what percentage of students sleep between 5 and 10 hours? (round to whole number)
About 84%
What is the End Behavior of the following graph? f1(x)=log2/3(x+2)-4
Asx→−2,f(x)→+∞ Asx→+∞,f(x)→−∞
(𝟐𝟏+𝟐𝟑𝒊)/(−𝟒−𝟏𝟎𝒊)
−𝟏𝟓𝟕/𝟓𝟖 + 𝟓𝟗i/𝟓8
𝟏𝟓−𝟏𝟏𝒊/−𝟖−𝟏𝟎𝒊
−𝟓/𝟖𝟐 + 𝟏𝟏𝟗i/𝟖𝟐
(𝟏𝟎 − 𝟖𝒊√𝟏𝟑)^2
-732-160i√13
Find the sum of the following infinite series, if it exists. 0.5 + 0.25 + 0.125 + ...
1
Find the average of a/3, a/6, a/9
11a/54
(𝟖 + 𝟏𝟐𝒊) + (𝟏𝟏𝟔 − 𝟑𝟎𝒊)
124-18i
Simplify √(4x^2*y^2*z^4)
2*|xy|*z^2
Which histogram represents the following data set? 45, 55, 25, 48, 36, 61, 52, 31, 8, 41, 58, 40, 55, 47, 60, 28, 44, 63, 18, 50, 57, 37, 16, 56, 40, 50
2, 2, 4, 6, 8, 4
Use a calculator to evaluate each expression to the nearest ten-thousandth. log 101
2.0043
The graph below represents the speed of a runner given the time. How long is the runner going less than 10 mph?
20 minutes
Which histogram represents the following data set? 31, 67, 8, 37, 12, 87, 14, 34, 105, 57, 42, 8, 16, 54, 17, 20, 72, 23, 27, 63, 24, 52, 14, 44, 27, 5, 28, 22, 33, 15, 6, 36, 41, 21, 46
4, 8, 7, 6, 4, 2, 2, 1, 0, 1
Let Q (in grams) represent the mass of a quantity of carbon-14, which has a half-life of 5730 years. The quantity present after t years is \mathrm Q(\mathrm t)=10\cdot\left(\frac12\right)^{\left(\frac{\mathrm t}{5730}\right)}Q(t)=10⋅(21)(5730t). Determine the quantity present after 2000 years.
7.851 grams
Ted buys wood to build his guitars. Find the number of blocks of mahogany that Ted can afford to buy if he wishes to spend a total of $5000 this month, mahogany costs $450 per block, and he has already bought 7 blocks of spruce at $200 each.
8
(𝟏𝟏 − 𝟒𝒊√𝟕)^𝟐
9-88i√𝟕
For a normal distribution, what percentage of data falls within three standard deviations of the mean?
99.7%
A researcher for an airline interviews all of the passengers on five randomly selected flights. Identify which sampling technique is used.
Cluster
Mike invested $93,000 at 8.19% compounded weekly. What will Mike's account balance be in 11 years?
Correct $228,786.85
Solve √(5y-3) = √(7y+9)
No solution
Absolute Value - The absolute value of a number is the distance the number is from zero on the number line. Asymptotes - An asymptote is a line or curve that approaches a given curve arbitrarily closely. A graph never crosses a vertical asymptote, but it may cross a horizontal or oblique asymptote. Base (of a Power) - The number or expression used as a factor for repeated multiplication. Common Logarithm - A logarithm with a base of 10. A common logarithm is the exponent, a, such that 10a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 102 = 100. Compound Interest Formula - A method of computing the interest, after a specified time, and adding the interest to the balance of the account. Interest can be computed as little as once a year to as many times as one would like. The formula is where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times compounded per year, and t is the number of years. Continuous Compound Interest Formula - Interest that is theoretically, computed and added to the balance of an account each instant. The formula is , where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, and t is the time in years. Decay Factor - The base number "b" with a value 0 < b < 1 in a function of the form where 0 < b < 1. Degree - The exponent of a number or expression. Degree of a Polynomial - The largest exponent of x which appears in the polynomial. Domain - The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation. Estimate - A guess about the size, cost, or quantity of something. Exponential - A number written with an exponent. For example, 63 is called an exponential expression. Exponential Function - A function of the form where a, h, and k are real numbers, b > 0, and a and b are not equal to 1. Exponential Decay Function - A function of the form where 0 < b < 1. Exponential Growth Function - A function of the form where b > 1. Factor - When two or more integers are multiplied, each integer is a factor of the product. "To factor" means to write the number or term as a product of its factors. Function - A rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set. Geometric Sequence - Is a sequence with a constant ratio between successive terms. Geometric Series - The expression formed by adding the terms of a geometric sequence. Graph of a Function - The set of all the points on a coordinate plane whose coordinates make the rule of function true. Growth Factor - The base number "b" with a value b > 1 in a function of the form. Integer - The set of numbers ...,-3,-2,-1,0,1,2,3,... Interest - The percent of the money on deposit (the principal) paid to a lender for the use of the principle. Interval - A regular distance or space between values. The set of points between two numbers. Natural Base e - Euler's number e with the approximation of 2.718... Natural Logarithm - A logarithm with a base of e. ln b is the exponent, a, such that . The natural logarithm of x is written ln x and represents . For example, ln 8 = 2.0794415...because . Pattern - A set of numbers or objects that are generated by following a specific rule. Power - The exponent of a number or expression, which indicates the number of times the number or expression is used as a factor. Polynomial - An algebraic expression involving variables with nonnegative integer exponents with one or more unlike terms. Quadratic Function - A function of degree 2 whose graph is a parabola. Range - The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs. The range is the output in a function or a relation. Rate - A comparison of two quantities that have different units of measure. Recursive - A type of sequence in which successive terms are generated by preceding terms in the sequence. Scatterplot - The graph of a collection of ordered pairs that allows an exploration of the relationship between the points. Substitute - To replace one element of a mathematical equation or expression with another. Sum of finite geometric series - The sum Sn, of the first n terms of a geometric sequence is given by , where a1 is the first term and r is the common ratio (r≠1). Sum of infinite geometric series - The general formula for the sum S of an infinite geometric series a1, a2, a3, a4,... with common ratio r where |r|<1 is . If an infinite geometric series has a sum, i.e. if |r| < 1, then the series is called a convergent geometric series. All other geometric (and arithmetic) series are divergent. Symmetry - A mirror image across a line such as the x axis, y axis or across the origin. Three-Dimensional Figure - Figures that have length, width, and height. Two-Dimensional Figure - Figures that have length and width (no height). Unit - A fixed amount that is used as a standard of measurement. Variable - A letter or symbol used to represent a number. x-intercept - The value on the x-axis where a graph crosses the x-axis. y-intercept - The value on the y-axis where a graph crosses the y-axis.
Math Modeling Key Terms
What is the End Behavior of the following Graph? f1(x)=-3*(2/5)^(x+2)-3
As x approaches -infinity, y approaches -infinity. As x approaches infinity, y approaches -3.
Which type of study contains a treatment?
Experiment
What is the shape of this histogram?
Unimodal (one clear peak)
If you are studying the effects of UV rays on eyesight and you group 10 people together and make them wear sunglasses for 10 weeks and see if their eyesight is affected and then take another group and do not give them sunglasses and test their vision after 10 weeks, what is the response variable? (note this is not an ethical study)
Vision Test
State the domain of the function graphed below.
{x/x≥−3}
Katie invested $33,750 at 11.17% compounded continuously. What will Katie's account balance be in 10 years?
$103,128.98
Jeremy is investing $47,000 in an account paying 9.26% interest compounded continuously. What will Jeremy's account balance be in 42 years?
$2,296,973.35
Jimmie invested $27,500 at 6.27% compounded continuously. What will Jimmie's account balance be in 50 years?
$632,187.49
An electrician earns $24 an hour for a regular work week (40) hours. She earns time and a half for overtime. The weekly wage function is... How much will she earn if she works 38.5 hours this week?
$924
(𝟏𝟐 − 𝟏𝟓𝒊) − (𝟒𝟖 − 𝟓𝟐𝒊)
-36+37i
Given a cubic function, what would be the maximum number changes of direction (turning points)?
0
Cameron Indoor Stadium at Duke University is one of the most revered sites in all of college basketball, as well as in all of sports period. Duke's men's and women's basketball programs have attained quite a few wins in the building over the last seventy years. Cameron Indoor Stadium is capable of seating 9,450 people. For each game, the amount of money that the Duke Blue Devils' athletic program brings in as revenue is a function of the number of people in attendance. If each ticket costs $45.50, find the domain and range of this function.
The domain is all integer values in the interval [0, 9450]. The range is all multiples of 45.5 in the interval [0, 429975].
(5+2i)+(3-7i)
8-5i
State the range of the function graphed below.
(negative infinity , 3]
What is the fifth number in the 7th row of Pascal's Triangle?
15
Graduate Management Aptitude Test (GMAT) scores are widely used by graduate schools of business as an entrance requirement. Suppose that in one particular year, the mean score for the GMAT was 473, with a standard deviation of 104. What values are three standard deviations within the mean?
161 and 758 (473-3(104) = 161; 473+3(104)=785)
Suppose a random sample of 84 men has a mean foot length of 26.9 cm with a standard deviation of 2.1 cm. What is an 95% confidence interval for this data?
26.451 to 27.349
For a normal distribution, what percentage of data falls within one standard deviation of the mean?
68%
The mean of a set of normally distributed data is 600 with a standard deviation of 20. What percent of the data is between 580 and 620?
68%
What allows us to apply normal calculations to non-normal distributions?
Central Limit Theorem
If you are studying the effects of UV rays on eyesight and you group 10 people together and make them wear sunglasses for 10 weeks and see if their eyesight is affected and then take another group and do not give them sunglasses and test their vision after 10 weeks, what is the focus of the study? (note this is not an ethical study)
Effects of UV rays on eyesight
If you are studying the effects of UV rays on eyesight and you group 10 people together and make them wear sunglasses for 10 weeks and see if their eyesight is affected and then take another group and do not give them sunglasses and test their vision after 10 weeks, what is the variable of interest? (note this is not an ethical study)
Eyesight
The graph of y=−2x2+4 is a right−side−up parabola.
False
What is a potential source of bias (if any) if you are surveying students to find out their opinion of a new teacher with the following question: Is Mr Wilson a terrible teacher?
Implies that Mr. Wilson is a bad teacher.
The spirit club is conducting a survey to find out the favorite sport of the students in the high school. Jake, a member of the spirit club asks the members of the basketball team what is their favorite sport. Is this sampling method representative of the entire school and explain?
No, the members of the basketball team are more likely to say that basketball is their favorite sport.
Best Buy wants to know if Rewards customers prefer Sony or Fuji cameras. They asked everyone at the sandwich shop next door what kind of camera they prefer. Is this type of sampling the best method of sampling in this scenario?
No, the people in the sandwich shop may not be Rewards customers, or even shop at Best Buy.
A biologist wants to determine the effect of a new fertilizer on tomato plants. What could be a response variable?
Plant Growth
A biologist wants to determine the effect of a new fertilizer on tomato plants. What would be the control?
Plants not treated with the Fertilizer.
Describe the shape of the distribution in the following data set. Then describe the center and spread of the data using either the mean and standard deviation or the five-number summary. 31, 67, 8, 37, 12, 87, 14, 34, 105, 57, 42, 8, 16, 54, 17, 20, 72, 23, 27, 63, 24, 52, 14, 44, 27, 5, 28, 22, 33, 15, 6, 36, 41, 21, 46
Positively skewed; the distribution is skewed, so use the five-number summary. The range is 5 to 105 pounds. The median weight is 28 pounds, and half of the dogs' weights are between 16 and 46 pounds.
If you are studying the effects of UV rays on eyesight and you group 10 people together and make them wear sunglasses for 10 weeks and see if their eyesight is affected and then take another group and do not give them sunglasses and test their vision after 10 weeks, what is the treatment? (note this is not an ethical study)
Sunglasses
A sample consists of every 25th student from a group of 1000 students. The sampling technique used is:
Systematic
There are two calculus classes at your school. Both classes have a class average of 75.5. The first class has a standard deviation of 10.2 and the second has a standard deviation of 22.5. How do the classes compare?
The first class' scores are closer together and the second's score are more spread out.
The local grocery store wants to know if the shoppers prefer Crest or Colgate toothpaste. The store gives all of its shoppers a value saver card, then uses a random number generator to select 30 card numbers and surveys those shoppers. Is this sampling method representative of the entire population and explain?
Yes, all shoppers have a value card and the shoppers interviewed were chosen by a random number generator
Given the graph below, find the equation.
y=x^2-5
Simplify 6√((m+4)^6)
|m+4|
The graph below shows the cost of your cellphone service based on the number of minutes you used your phone. What is the cost if you used 360 minutes?
$60
(-3-7i)+(4+2i)
1-5i
(6+3i)/(7-5i)
27/74 + 51i/74
A new toy hits the local store. Sales (in hundreds) increase a steady rate for several months, then decrease at about the same rate. This can be modeled by the function S(m)=-0.625\left|\mathrm{m}-8\right|+5S(m)=−0.625∣m−8∣+5 In what month(s) were 250 toys sold?
4 and 12
Mai is kayaking on a river that has a current of 2 miles per hour. If r represents her rate in calm water, then (r + 2) represents her rate with the current, and (r - 2) represents her rate against the current. Mai kayaks 2 miles downstream and then back to her starting point. Use the formula for time, t=\frac{\mathrm d}{\mathrm r}t=rd, where d is the distance, to write a simplified expression for the total time it takes Mai to complete the trip.
4r/(r+2)(r-2) hours
When sending a rectangular package through the U.S. Postal Service, the combined length and girth (perimeter of the cross section) cannot exceed 108 inches. Up to what length can a package be if the width and height are 10 and 15 inches?
58 inches
The zero in the following equation with a multiplicity of 2 is ... y=3x(x−4)3(x+6)2(2x+1)
6
(𝟏𝟐 − 𝟗𝒊)^2
63-216i
A new toy hits the local store. Sales (in hundreds) increase at a steady rate for several months, then decrease at about the same rate. This can be modeled by the function S(m)=-0.625\left|\mathrm{m}-8\right|+5S(m)=−0.625∣m−8∣+5 In what month(s) were 400 toys sold?
7th and 10th
Radical Functions Key Terms
Algebra - The branch of mathematics that deals with relationships between numbers, utilizing letters and other symbols to represent specific sets of numbers, or to describe a pattern of relationships between numbers. Coefficient - A number multiplied by a variable. Equation - A number sentence that contains an equality symbol. Expression - A mathematical phrase involving at least one variable and sometimes numbers and operation symbols. Extraneous Solution(s) - A solution of the simplified form of the equation that does not satisfy the original equation. Index of a Radical - The " n " in the nth root of a, written as . Inequality - Any mathematical sentence that contains the symbols > (greater than), < (less than), ≤ (less than or equal to), or ≥ (greater than or equal to). Polynomial - A mathematical expression involving the sum of terms made up of variables to nonnegative integer powers and real-valued coefficients. Radical Function - A function containing a root. The most common radical functions are the square root and cube root functions, Reciprocal - Two numbers whose product is one. For example, . Variable - A letter or symbol used to represent a number.
What is the End Behavior of the following Graph? f1(x)=4*(2/5)^(x+3)-4
As x approaches -infinity, y approaches infinity. As x approaches infinity, y approaches -4.
What is the End Behavior of the following Graph? f1(x)=-3*2^(x-5)-4
As x approaches infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches -4.
State the domain and range of the function graphed below.
Domain: (-7 , infinity) Range: (negative infinity , 3)
What is the Domain and Range of the following Graph? f1(x)=4*(2/5)^(x+3)-4
Domain: (-infinity, infinity) Range: (-4, infinity)
What is the Domain and Range of the following Graph? f1(x)=-3*2^(x-5)-4
Domain: (-infinity, infinity) Range: (-infinity, -4)
What is the Domain and Range of the following graph?
Domain: (1, infinity) Range: (- infinity, + infinity)
What is the Domain and Range of the following: y=1.5(2)^x
Domain: All Real #s Range: y>0
Determine the domain and range for the given function. y=∣x−1∣+3
Domain: All Real Numbers Range: All Real Numbers greater than or equal to 3
What is the Domain and Range of the following: y=4(3)^x
Domain: All Real Numbers Range: y>0
State the domain and range of the function graphed below.
Domain: [-7/3 , infinity) Range: [-5 , infinity)
Exponential Function - A function of the form are real numbers, b > 0, and a and b are ≠ 1. Exponential Growth Function - A function of the form where b > 1. Growth Factor - The base number "b" with a value b > 1 in a function of the form where b > 1. Asymptote - An asymptote is a line or curve that approaches a given curve arbitrarily closely. A graph never crosses a vertical asymptote, but it may cross a horizontal or oblique asymptote. Exponential Decay Function - A function of the form where 0 < b < 1. Decay Factor - The base number "b" with a value 0 < b < 1 in a function of the form where 0 < b < 1. Natural Base e - Euler's number e with the approximation of 2.718... Common Logarithm - A logarithm with a base of 10. A common logarithm is the exponent, a, such that . The common logarithm of x is written log x. For example, log 100 = 2 because . Natural Logarithm - A logarithm with a base of e. lnb is the exponent, a, such that . The natural logarithm of x is written lnx and represents . For example, ln 8 = 2.0794415... because . Compound Interest Formula - A method of computing the interest, after a specified time, and adding the interest to the balance of the account. Interest can be computed as little as once a year to as many times as one would like. The formula is where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times compounded per year, and t is the number of years. Continuous Compound Interest Formula - Interest that is, theoretically, computed and added to the balance of an account each instant. The formula is , where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, and t is the time in years.
Exponential and Logarithm Key Terms
A polynomial function with a degree of 0 is called a Linear Function.
False
The domain of the graph of this equation is (7,∞). y=x3−2x2+5x+7
False
The domain of the graph of this equation is (−3.0249,+∞). y=-x^3+10x^2+x-3
False
End Behavior
Functions with a domain of all real numbers do not have a beginning or an end. Functions with "even degrees" do have an absolute maximum or minimum, while the "opposite" end of these functions will go to positive OR negative infinity. While functions with "odd numbered degrees" will go to negative infinity AND positive infinity at opposite ends.
Degree 0 1 2 3 4 5
Name Constant Linear Quadratic Cubic Quartic Quintic
Which functions have graphs that are symmetrical with respect to the origin?
Odd
Quadratics Revisited Key Terms
Polynomial - The sum or difference of two or more monomials. Constant - A term with degree 0 (a number alone, with no variable). Monomial - An algebraic expression that is a constant, a variable, or a product of a constant and one or more variables (also called "terms"). Binomial - The sum or difference of two monomials. Trinomial - The sum or difference of three monomials. Integers - Positive, negative and zero whole numbers (no fractions or decimals). Like Terms - Terms having the exact same variable(s) and exponent(s). Coefficient - Number factor; number in front of the variable. Imaginary Number - A number that involves i which is Complex Number - A number with both a real and an imaginary part, in the form a + bi Conjugate - The same binomial expression with the opposite sign. Greatest Common Factor - Largest expression that will go into the terms evenly. Zeros - The roots of a function, also called solutions or x-intercepts. Linear - A 1st power polynomial. Quadratic - A 2nd power polynomial. Cubic - A 3rd power polynomial. Quartic - A 4th power polynomial. Intercepts - Points where a graph crosses an axis. System of Equations - n equations with n variables. Point of Intersection - The point(s) where the graphs cross. Consistent - Has at least one solution. Inconsistent - Has no solution. Domain - The values for the x-variable. Range - The values for the y-variable. Extrema - Maximums and minimums of a graph. Rational Exponent - A rational number written in the exponent of the form , where a is the base of the exponent, m is the numerator (power), and n is the denominator (root of the radical).
Operations with Polynomials Key Terms
Polynomial - The sum or difference of two or more monomials. Constant - A term with degree 0 (a number alone, with no variable). Monomial - An algebraic expression that is a constant, a variable, or a product of a constant and one or more variables (also called "terms"). Binomial - The sum or difference of two monomials. Trinomial - The sum or difference of three monomials. Integers - Positive, negative and zero whole numbers (no fractions or decimals). Like Terms - Terms having the exact same variable(s) and exponent(s). Coefficient - Number factor; number in front of the variable. Linear - A 1st power polynomial. Quadratic - A 2nd power polynomial. Cubic - A 3rd power polynomial. Quartic - A 4th power polynomial. Pascal's Triangle - A number triangle with numbers arranged in staggered rows such that , where is a binomial coefficient. Synthetic Division - Is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor - and it only works in this case. Long Division - Is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. Compositions of Functions - A composition of a function is the point wise application of one function to the result of another to produce a third function. Intuitively, composing two functions is a chaining process in which the output of the inner function becomes the input of the outer function. Inverses of Functions - An inverse of a function is a function that undoes the action of another function. A function g is the inverse of a function f if whenever y = f(x) then x = g(y). In other words, applying f and then g is the same thing as doing nothing. We can write this in terms of the composition of f and g as g(f(x)) = x. A function f has an inverse function only if for every y in its range there is only one value of x in its domain for which f(x) = y. This inverse function is unique and is frequently denoted by f-1 and called "f inverse."
Polynomial - The sum or difference of two or more monomials. Constant - A term with degree 0 (a number alone, with no variable). Monomial - An algebraic expression that is a constant, a variable, or a product of a constant and one or more variables (also called "terms"). Binomial - The sum or difference of two monomials. Trinomial - The sum or difference of three monomials. Degree of the Polynomial - The largest sum of the exponents of one term in the polynomial. Integers - Positive, negative and zero whole numbers (no fractions or decimals). Like Terms - Terms having the exact same variable(s) and exponent(s). Coefficient - Number factor; number in front of the variable. Imaginary Number - A number that involves i which is . Complex Number - A number with both a real and an imaginary part, in the form . Conjugate - The same binomial expression with the opposite sign. Greatest Common Factor - Largest expression that will go into the terms evenly. Zeros - The roots of a function, also called solutions or x-intercepts. Constant - A "0" power (degree) polynomial. Linear - A 1st power (degree) polynomial. Quadratic - A 2nd power (degree) polynomial. Cubic - A 3rd power (degree) polynomial. Quartic - A 4th power (degree) polynomial. Quintic - A 5th power (degree) polynomial. Intercepts - Points where a graph crosses an axis. System of Equations - n equations with n variables. Point of Intersection - The point(s) where the graphs cross. Consistent - Has at least one solution. Inconsistent - Has no solution. Domain - The values for the x variable. Range - The values for the y variable. Extrema - Maximums and minimums of a graph.
Polynomial Functions Key Terms
Rational Functions Key Terms
Rational Expression - An expression that can be written as a fraction. Excluded Values - Values that make the expression undefined (0 in the denominator). Like Terms - Terms having the exact same variable(s) and exponent(s). Extraneous Solutions - Solutions that make the expression undefined. Intercepts - Points where a graph crosses an axis. Domain - The values for the x-variable. Range - The values for the y-variable. Zeros - The roots of a function, also called solutions or x-intercepts. Asymptotes - Vertical and horizontal lines where the function is undefined. Extrema - Maximums and minimums of a graph. End Behavior - The rise or fall of the ends of the graph. Conjugate - The same binomial expression with the opposite sign. Greatest Common Factor - Largest expression that will go into the terms evenly. Lowest Common Denominator - Denominator that is the smallest multiple of all of the denominators.
The graph below represents the speed of a runner given the time. What is happening between 10 and 20 minutes?
The runner is steadily slowing from 10 mph to 5 mph.
The graph of x^4-5x^3+5x^2-2x4−5x3+5x2−2 is "W" shaped.
True
The range of the graph of this equation is [2,∞). y=x^4-4x^3+\frac{14}{3}x^2+2
True
Using the graph below, find f(10).
Undefined
Given that the loudness L of a sound in decibels is given by is L\;=\;10\;log_{10}\;RL=10log10R, where R is the relative intensity. If the intensity of a certain sound is tripled, by how many decibels does the sound increase?
about 4.8 dB
What is the equation of the following graph?
f(x)=2⋅log7(x)
Imaginary Numbers Basics
i = square root of -1 i^0 = 1 i^1 = i i^2 = -1 i^3 = -i i^4 = 1 i^5 = i i^6 = -1 i^7 = -i
Write the function for the graph shown.
y = -|x| + 3
A college with a graduating class of 4000 students in the year 2010 predicts that its graduating class will grow 5% per year. Write an exponential function to model the number of students y in the graduating class t years after 2010.
y=4000(1.05)^t
What is the equation/inequality of the following graph?
y≤−√(x+5)−4
The points (63, 121), (71, 180), (67, 140), (65, 108), and (72, 165) give the weight in pounds as a function of height in inches for 5 students in a class. Give the points for these students that represent height as a function of weight.
{(121, 63), (180, 71), (140, 67), (108, 65), (165, 72)}