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What does ƒ(x) mean? Think of the function notation as a replacement for y. It reads f of x.ƒ(x) = 2x + 1 is also known as y = 2x + 1.ƒ(x) = |-x + 5| is also known as y = |-x + 5|.ƒ(x) = 5x2 + 3x - 10 is also known as y = 5x2 + 3x - 10.What do these variations of notation share? ƒ(t) = -2t2ƒ(b) = 3ebƒ(p) = 10p + 12Whether the function begins with ƒ(x) or ƒ(t) or ƒ(b) or ƒ(p) or ƒ(♣), it means that the outcome of ƒ depends on whats in the parentheses.ƒ(x) = 2x + 1 (The value of ƒ(x) depends on the value of x.)ƒ(b) = 3eb (The value of ƒ(b) depends on the value of b.)Learn how to use a graph to find specific values of ƒ. What is ƒ(2)?In other words, when x = 2, what is ƒ(x)?Trace the line with your finger until you get to the part of the line where x = 2. What is the value of ƒ(x)?Answer: 11 What is ƒ(-3)?In other words, when x = -3, what is ƒ(x)?Trace the graph of the absolute value function with your finger until youre touching the point where x = -3. What is the value of ƒ(x)?Answer: 15 What is ƒ(-6)?In other words, when x = -6, what is ƒ(x)?Trace the parabola with your finger until you touch the point at which x = -6. What is the value of ƒ(x)?Answer: -18 What is ƒ(1)?In other words, when x = 1, what is ƒ(x)?Trace the exponential growth function with your finger until you touch the point at which x = 1. What is the value of ƒ(x)?Answer: 3 What is ƒ(90°)?In other words, when x = 90°, what is ƒ(x)?Trace the sine function with your finger until you touch the point at which x = 90°. What is the value of ƒ(x)?Answer: 1 What is ƒ(180°)?In other words, when x = 180°, what is ƒ(x)?Trace the cosine function with your finger until you touch the point at which x = 180°. What is the value of ƒ(x)?Answer: -1 How to Calculate a Demand Function Tools of Economics The 45-Degree Line of Economics... An Explanation of the Supply &... The Advantages of Purchasing Power... What Is the Difference Between... How to Calculate Residual Variance What Does the Marginal Rate of... Equations Used in Economics How to Calculate PED How to Calculate a Demand Function... How to Calculate 95% Confidence... How do I Calculate Seasonality? How to Calculate the Slope of a... How to Calculate the Amount of... The Use of Models in Economics Fixed-Proportions Production Functions Difference Between Complementary... One goal of statistics is to present data in a meaningful way. Often, data sets involve millions (if not billions) of values. This is far too many to print out in a journal article or sidebar of a magazine story. Thats where graphs can be invaluable, allowing statisticians to provide a visual interpretation of complex numerical stories. Seven types of graphs are commonly used in statistics. Good graphs convey information quickly and easily to the user. Graphs highlight the salient features of the data. They can show relationships that are not obvious from studying a list of numbers. They can also provide a convenient way to compare different sets of data.Different situations call for different types of graphs, and it helps to have a good knowledge of what types are available. The type of data often determines what graph is appropriate to use. Qualitative data, quantitative data, and paired data each use different types of graphs. Erik Dreyer Getty ImagesA Pareto diagram or bar graph is a way to visually represent qualitative data. Data is displayed either horizontally or vertically and allows viewers to compare items, such as amounts, characteristics, times, and frequency. The bars are arranged in order of frequency, so more important categories are emphasized. By looking at all the bars, it is easy to tell at a glance which categories in a set of data dominate the others. Bar graphs can be either single, stacked, or grouped.Vilfredo Pareto (1848–1923) developed the bar graph when he sought to give economic decision-making a more human face by plotting data on graph paper, with income on one axis and the number of people at different income levels on the other. The results were striking: They showed dramatically the disparity between rich and poor in each era over the course of centuries. Walker and Walker Getty ImagesAnother common way to represent data graphically is a pie chart. It gets its name from the way it looks, just like a circular pie that has been cut into several slices. This kind of graph is helpful when graphing qualitative data, where the information describes a trait or attribute and is not numerical. Each slice of pie represents a different category, and each trait corresponds to a different slice of the pie; some slices usually noticeably larger than others. By looking at all of the pie pieces, you can compare how much of the data fits in each category, or slice. Qwfp Wikimedia Commons CC BY 3.0A histogram in another kind of graph that uses bars in its display. This type of graph is used with quantitative data. Ranges of values, called classes, are listed at the bottom, and the classes with greater frequencies have taller bars.A histogram often looks similar to a bar graph, but they are different because of the level of measurement of the data. Bar graphs measure the frequency of categorical data. A categorical variable is one that has two or more categories, such as gender or hair color. Histograms, by contrast, are used for data that involve ordinal variables, or things that are not easily quantified, like feelings or opinions. A stem and leaf plot breaks each value of a quantitative data set into two pieces: a stem, typically for the highest place value, and a leaf for the other place values. It provides a way to list all data values in a compact form. For example, if you are using this graph to review student test scores of 84, 65, 78, 75, 89, 90, 88, 83, 72, 91, and 90, the stems would be 6, 7, 8, and 9, corresponding to the tens place of the data. The leaves—the numbers to the right of a solid line—would be 0, 0, 1 next to the 9; 3, 4, 8, 9 next to the 8; 2, 5, 8 next to the 7; and, 2 next to the 6.This would show you that four students scored in the 90th percentile, three students in the 80th percentile, two in the 70th, and only one in the 60th. Youd even be able to see how well students in each percentile performed, making this a good graph to understand how well students comprehend the material. ProdunisWikimedia CommonsPublic DomainA dot plot is a hybrid between a histogram and a stem and leaf plot. Each quantitative data value becomes a dot or point that is placed above the appropriate class values. Where histograms use rectangles—or bars—these graphs use dots, which are then joined together with a simple line, says statisticshowto.com. Dot plots provide a good way to compare how long it takes a group of six or seven individuals to make breakfast, for example, or to show the percentage of people in various countries who have access to electricity, according to MathIsFun. Illia Connell Wikimedia Commons CC BY 3.0A scatterplot displays data that is paired by using a horizontal axis (the x-axis), and a vertical axis (the y-axis). The statistical tools of correlation and regression are then used to show trends on the scatterplot. A scatterplot usually looks like a line or curve moving up or down from left to right along the graph with points scattered along the line. The scatterplot helps you uncover more information about any data set, including:The overall trend among variables (You can quickly see if the trend is upward or downward.)Any outliers from the overall trend.The shape of any trend.The strength of any trend. Peter James Eaton Wikimedia Commons CC BY 4.0A time-series graph displays data at different points in time, so it is another kind of graph to be used for certain kinds of paired data. As the name implies, this type of graph measures trends over time, but the timeframe can be minutes, hours, days, months, years, decades, or centuries. For example, you might use this type of graph to plot the population of the United States over the course of a century. The y-axis would list the growing population, while the x-axis would list the years, such as 1900, 1950, 2000.
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