Course:MATH110/Archive/2010-2011/003/Groups/Group 04/Basic Skills Project
For the basic skills project, group 4 plans to outline the properties of functions in a detailed fashion.
What are functions?
Functions in math are when two variables are considered and put into an equation, and how changes in these variables affect each other. The two variables are either the domain or the range.
For example:
Measuring the amount of rainfall (range, or the y-value) in a week (domain, or the x-value)
Example of the function: y=2x+1
It is important to note that there are many different types of functions, and they depend on what degree they are. For example, if they are degree 1, they are linear functions. If they are degree 2, they are quadratic functions, degree 3 cubic functions and so on
GRAPHING FUNCTIONS There are many methods to graph functions. For quadratic functions, you can graph them by finding the vertex. For linear functions, by finding the slope and intercepts.
One of the most common functions is a linear function, so we're gonna start off with how to graph a linear function.
Graphing linear functions, for the most part, are simple.
Let's use the function: y=2x+1
To graph this linear function, we need to find two points. But we can't just use any two points. We will draw the graph by finding the x-intercept and the y-intercept.
To do this is fairly simple.
To find the x-intercept, you have to substitute y for zero, and to find the y-intercept, you have to substitute x for zero
FINDING THE X-INTERCEPT
y=2x+1 0=2x+1 -1=2x x=-1/2 - x-intercept
FINDING THE Y-INTERCEPT y=2x+1 y=2(0)+1 y=1 - y-intercept
What we do now is simple. Plot the two points of the intercept. Coordinates of x-intercept: (-1/2,0) Coordinates of y0intercept: (0,1)
Then simply draw a straight line through these two points, and that's the graph of the function!
For a more visual guide, refer to the video:
http://www.youtube.com/watch?v=mxBoni8N70Y
To draw quadratic functions may be a more difficult task
Although it's a little more complicated, there is a step by step process, and if you follow it, there should be no trouble at all.
To illustrate how to draw a quadratic function, we have to understand what a quadratic function is.
The standard format of a quadratic function is: y=ax^2+bx+c
a, b and c are integers. A cannot equal zero.
Step by step guide to how to graph a quadratic function:
1) Determine concavity 2)Find vertex 3) Find x-intercepts (if any) and y-intercepts
From here, we can use an example to demonstrate how to graph a quadratic function.
Let's use the function: y=x^2+4x+4
So first, we have to find the concavity. In this case, we know that it is concave up because the a value is one, which is positive meaning that it is concave up.
Second, we have to find the vertex. To do this we use the formula:
-b/2a
This formula gives us the x-value of the vertex, which means we can plug in that value to find the y-value
So in this case, -b=-4 and 2a=2
So it is -4/2, which is -2
The x-value of the vertex is -2.
To find the y-value, we simply plug in -2 for x
So, y=(-2)^2+4(-2)+4 y=0
So the coordinates of the vertex is (-2,0)
Afterwards, we find the x and y intercepts. To do that, we can use the same method we used in determining the x and y intercepts for linear functions. We substitute x for zero if we want to find the y-intercept, and y for zero when we want to find the x-intercept.
The x-intercept: 0=x^2+4x+4
0=(x+2)^2 x=-2
The y-intercept y=(0)^2+4(0)+4 y=4
With all the information we have now, which includes the vertex (-2,0), the x-intercept (-2,0) and the y-intercept (0,4) we can draw the graph.
For a more visual demonstration, refer to the video:
http://www.youtube.com/watch?v=mDwN1SqnMRU
HOW TO USE THE GRAPH OF A FUNCTION A graph of a function is simply a graphical representation of the said function. It can be used as a visual aid into discerning the behavior of the two variables the function is comparing.
For example, let's say that a function is comparing the growth of a certain bacteria in Mars over time. This bacteria has a constant rate of growth, so it is a linear function. The x-axis of the graph will tell us the time, and the y-axis will tell us the growth of the bacteria. The graph will show us how much, visually, the bacteria has grown over a given period of time.
In other instances, graphs of functions can show us the height of a ball after it was thrown, or the displacement of a swing on a playground. Essentially, they are visual aids that are used to help us better understand the things that we are trying to compare.
WHAT IS THE DOMAIN OF A FUNCTION
What is the domain? The domain is what x can be, including all real numbers, that will keep the function real. The best way to show this is through an example
Consider the function: y=1/x
The x-value in the function can be any number, and the function will still be real. However, if x was 0, there would be an error and it would no longer be a function. In fact, 1/0 is impossible.
HOW TO FIND THE DOMAIN OF A FUNCTION Finding the domain of a function is actually quite simple. All we need to remember are two rules:
1) The denominator of a fraction in a function cannot be 0
2) A negative number cannot be under a root in a function
So basically, all x-values that make the denominator 0 is not part of the domain, and any x-value that makes a negative number under the root is not part of the domain
For a more visual guide, refer to the video: