Linear comes from the Latin word linearis, which means “created by lines”.
If a graph is linear it will form a straight.
Linear expressions consist of only one variable, and no exponents.
Y=25X+104 ← Linear
Y=4(3/5)ⁿ ← Non Linear, because of the variable as an exponent
Linear Non Linear (Quadratic)
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Interpreting a graph, is just a fancy way of saying, “how do you read a graph”. In other words what information does the graph tell us. Graphs contain much more information than most people think . Lets take a look at an example, to see how to interpret or read a graph. Click on the picture to see the example. The BIG Book of Graphs
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Since 1990 the price of Hockey tickets have been increasing consistently. Let P represents the price of the ticket and T represent years since 1990 ( so T=2 represents 1992 ). The price of tickets are displayed on the graph below. Since the price of tickets depends on the year, we say the price(p) depends on the year(t). For the majority of graphs the dependent variable will be matched with the vertical axis, and the independent variable matched with the horizontal axis. Our example is considered an increasing function because the P value goes up as you move from left to right.
A linear equation is an expression that makes a linear graph. Duh! There is more to it than that though, so lets look at the basic linear equation more in depth.
Most Linear equations are in slope intercept form, Y=mx+b
Slope intercept form is the easiest linear equation to work with and it provides us with a lot of valuable information about our function. Click on each variable to find out more.
Y= M x+ B
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The m value of our equation represents the slope of our line. Now what in the world is slope? Slope refers to the rate of incline or decline. It could also be said that slope is the relative steepness of the line For example, on a ski hill the slope of a black diamond is bound to be greater than that of the bunny hill. As you read a graph from left to right, the slope refers to the change in the Y value over the change in the X value. Consider the graph to the right and the two points P 1 and P 2. The slope of this line would be expressed as M=Y 2 -Y 1 / X 2 - X 1 Click here for more practice with slope
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M=Y 2 - Y 1 / X 2 - X 1 is a fancy way of saying that the slope is equal to rise/run. Now Its Your Turn Assume that the points (1,2) and (5,4) are both on the same line. What is the slope of this line? Click on the video below for help Click here for the Answer
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M= Y 2 - Y 1 / X 2 - X 1 M=4 – 2 / 5 -1 = 2 / 4 M= 1/ 2 This means that for every one value of Y you increase, you go right two X values. Equation for Slope Inserting (1,2) and (5,4) Simplify M=1/2 is an Increasing slope, a slope that is a negative number is said to be a decreasing slope because the line goes down from left to right. Click Here to go back to “What Is A Linear Function”
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The B value of our equation represents the Y intercept of our line. The Y intercept is the Y value where the line crosses the axis. The X value is always 0 at the Y intercept. Y= 3/2X + 2 -> Y intercept = 2 Y= 3/2X – 6 -> Y intercept = -6 The line in the graph to the left has a y intercept of 6, therefore the B value is also 6.
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Finally we can graph linear equations! Here we will use our knowledge of equations and their properties to construct a table that will depict data for our equation. Lets look at an example. Lets consider the equation Y= -3X+2. We shall choose several values of X and compute their Y values to fill out our table. Lets choose -1, 0, 1, 2. -3(-1)+2=5 -3(0)+2=2 -3(1)+2=-1 -3(2)+2=-4 Once we have compute the values we can arrange them is a table like the one below Click here to see how our table turns into a graph! X value Y value -1 5 0 2 1 -1 2 -4
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Y= -3X+2 -> ( X, Y ) -> ( -1, 5 ) -> ( 0, 2 ) -> ( 1, -1 ) -> ( 2, -4 ) Notice that the table allows us to pull points out that we will use to graph. All that is left now is to plot these points on a graph. Congrats on graphing your first equation! Hopefully it is the first of many. Notice the graph is decreasing because of the negative slope, and the Y intercept of 2. These are things that we could have concluded by examining our m and b values. X value Y value -1 5 0 2 1 -1 2 -4
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Math is a unique subject, in that there is always more than one way to approach a problem. Linear Equations are no different. In addition to using a table, equations can be graphed using the equations themselves Remember that Y=M(X)+B tells us the slope and the Y intercept. The y intercept is a point in itself and to complete our graph we only need one additional point. To attain our additional point we simply use the slope. It might be useful, if you haven't already to review “What Is A Linear Equation.” Lets take the example of, Y= 3/5X+4. Click here to view the graph and solution to this problem.
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Y=M(X)+B Y=3/5X+4 M=3/5 B=4 Slope is, rise (3) / run (5) Y intercept (0,4)
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So what is the point of knowing all this mathematical jargon. Many real life situations are modeled in linear fashions. Graphs and tables cram a lot of information into a single page. They allow the reader to predict and compare different models. Graphs can even be extended to predict information outside of the know data. A runner may use a linear model to predict the miles he will have ran after a certain number of days. Employees may use linear models to compare the salaries of certain jobs.
Practice finding slope, Y intercept, and graphing.
Application Problems
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Given the points (2, 5) and (4, 12), find an equation whose line includes these points. A. What is the slope of this line? B. What is the Y intercept? C. Graph your equation D. Is function increasing or decreasing? Answer
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In December the Grand Rapids Symphony Orchestra opened their newest performance, entitled The Music of Charles Ives. The show was very popular for the first couple of months. Recently though the popularity of the show has fallen off. The performers will discontinue the act is they sell less than 1,700 tickets per month. The table below shows the months and the amount of tickets sold. 1. Use your knowledge of linear equations to graph the data in the table? 2. Should the act continue to play in June? Support your answer. Answer Month Tickets Sold December 10500 January 9000 February 7500 March 6000 April 4500 May 3000
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2. Considering they do not want to perform unless they sell 1,700 tickets, I would recommend the GRSO discontinue their act. Their ticket sales are decreasing, and are expected to be at only 1,500 in June.