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Linear Functions Explained
Transcript of Linear Functions Explained
How are linear functions graphed?
What do the parts of a linear function mean?
How can you write a linear function?
What makes something a linear function?
How do you find points on a function?
What are the many properties of a function?
and much more. So just what is a linear function? A linear function is a function that shows a linear relationship between two variables. Wait, what does that mean exactly? A Linear Function is usually "spelled out" in the following form, y=mx+b. m and b can be any real number that you can think of, you could have y=x+50 (with m being 1) or you could have y=1/682x+3, any numbers will work. Because x is always multiplied by the same number all of the time, this function will always graph a line, making this a LINEar function. The number that you multiply x by (better known as "m") determines the slope on a graph, or how steep the line is. Whether or not the number is negative will determine the direction that it is traveling as well. Here you will find all that you need to know about linear functions and how they are used, you can also find answers to many of your questions about linear functions such as... There are also many other forms of a linear function that you can use to describe the same thing, these are: general form, slope intecept form, x-y intercept form, point slope form, two point form, and intercept form. This is the most common way of writing a linear equation:
This is called slope intercept form. So how do we write a linear equation or function? To write the general form of a linear function you write it in the form of ax+by+c=0. To write an equation in x-y intercept form you must write it as ax+by=c. This form is very similar to general form, however you can use this form to easily find the x and y intercepts of the graph. To find the x intercept or the zero of y simply divide c by a, to find the y intercept or the zero of x, just divide c by b. Point slope form is written as y-y1=m(x-x1), with y1 and x1 being parts of an ordered pair on the graph of the equation. This form of a linear function is good for finding the equation of a line when all you know about it is one set of ordered pairs and the slope. To find the equation, you put the slope in place of m, then with the point (x1, y1), put x1 and y1 in their place on the equation. One of the most common mistakes that people make when they write this equation is that they forget to write y-y1 and x-x1, instead they sometimes put a + sign instead of a - sign. Don't forget that or else your equation won't work. Two point form is similar to point slope form but it uses the (x1, y1) and (x2, y2) points to represent the slope. To do this you must put y1-y1 over x1-x2. This gives you the slope. Therefore, the equation is written as y-y1=((y1-y2)/(x1-x2))(x-x1) This works because the y intercept occurs when x equals zero (and vice-versa), therefore, when x= zero the only part of the equation left is by=c so you can just divide c by b to get the y intercept. The same method works for finding the x intercept with a and c, just divide c by the number before x to find x intercept and divide by the number before y to find the y intercept. How you graph a linear function is dependant upon a few things including the numbers used in the function and how the function is written. In general form and x-y intercept form, you can find the individual intercepts of the graph and use a straight edge to draw a line along those two points. In the slope intercept form, you can graph the y intercept b and then use the slope to find the remaining graph, simply using rise over run to gradually add to the graph. For example, if the slope in an equation is equal to 2, then you simply convert to rise over run, which is 2/1, then simply go up 2 and over 1 and graph a single point. Repeat this until you have a complete graph. Intercept form is written as x/a+y/b=1, in this form, a is the x intercept and b is the y intercept. In point slope form, the easiest method of finding the graph of the function is to graph the point within the equation, (x1, y1) and use the slope to continue the graph in the same way as what you did with slope intercept form. For the two point form of the equation, you can find both of the points in the equation (x1, y1) and (x2, y2) and graph them to give you two points that you can them draw a straight line across with a straight edge like in x-y intercept form. To find the graph of an equation in intercept form, you graph the x and y intercepts a and b then use a straight edge to connect the two and finish the graph. With some of these graphs, the points that you use to draw a srtaight edge on might end up being a fraction, this might make your graph less accurate because it would be harder to graph. Y X This is a graph of y=2x-2
This same line can also be written in other forms besides y=2x-2, the other forms are:
general form is 2x-y+(-2)=0
xy intercept form is 2x-y=2
point slope form is y-0=2(x-1)
two point form is y-0=((0-(-2))/(1-0)(x-1)
and intercept form is x/1+y/(-2)=1 These are all different ways of writing the same equation and the same graph and line, just in different ways. It is up to you to find out which one works best for you. For this type of equation, simply measure the distance from one point to another and use the results to find the slope of the graph instead of drawing across the two points. Another method would be to convert the equation from one form to another, like from xy intercept form to slope intercept form. Real life examples of linear function usually involve measurements and conversions for money, time, weights, and volume. These functions are vital to providing good measurements when cooking, planning a trip, or building almost anything. Sugar cookie recipe: doubling the recipe:
sugar 1 1/4 cups x2 =2 1/2 cups
flour 2 1/2 cups x2 =5 cups
butter 1 cup x2 =2 cups
eggs 2 x2 =4 eggs
vanilla extract 1 tsp x2 =2 tsp
baking soda 1 tsp x2 =2 tsp
Mix all ingredients and bake for 10 minutes at 350*F Using the function y=2x to get enough ingredients for double the original cookie recipe
Functions are also used to help people figure things out financially. If a cell phone company requires that you pay them 5 dollars when you start their service and another 10 dollars per month after that.
This can be shown as y=$10x+$5 With all of the different uses for linear functions they are the most versatile function that you can find, applicable to almost any situation from driving to work to flying the space shuttle.
The possibilities are endless. So wherever you go or whatever you do, never forget what a function can do for you.