Showing posts with label graphing. Show all posts
Showing posts with label graphing. Show all posts

Thursday, April 23, 2015

Graph Lines Using Intercepts

The x-intercept is the point where the graph intersects the x-axis and the y-intercept is the point where the graph intersects the y-axis. These points have the form (x, 0) and (0, y) respectively.
x and y-intercepts
To find the x- and y-intercepts algebraically, we use the fact that all x-intercepts have a y-value of zero and all y-intercepts have an x-value of zero.  For example,
Graph:  3x − 5y = 15
Tip 1: To find the y-intercept, set x = 0 and determine the corresponding y-value.  Similarly, to find the x-intercept we set y = 0 and determine the corresponding x-value.
 
Keep in mind that the intercepts are ordered pairs and not numbers.  In other words, the x-intercept is not x = 5 but rather (5, 0).

Two points determine a line. If we find the x- and y-intercepts, then we can use them to graph the line. As you can see, they are fairly easy to find. Plot the points and draw a line through them with a straightedge.
Done. Let’s do another one.
Graph: yx + 9
We begin by finding the x-intercept.
The x-intercept is (3, 0).
The y-intercept is (0, 9). Now graph the two points.
Graph of the Line
Tip 2: Use Desmos.com to check your answer – it’s totally free.  Just type in the equation.

This is a nice and easy method for determining the two points you need for graphing a line.  In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.

Graph −4x + 3y = 12 using the intercepts.

  

Graph −4x + 2y = −6 using the intercepts.

  

Graph  y = −5x +15 using the intercepts.

   

Graph  y = −3/4 x + 9 using the intercepts.


This brings us to one of the most popular questions in linear graphing.  Do all lines have x- and y-intercepts?  The answer is NO.  Horizontal lines, of the form y = b, do not necessarily have x-intercepts.  Vertical lines, of the form x = a, do not necessarily have y-intercepts.

Graph y = 3.

  

Graph x = −2.

Many students this method, but I will tell you, there is a better way. Even less work... [ Graph Lines using Slope and Intercepts ] Read on!
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The Rectangular Coordinate System

The rectangular coordinate system consists of two real number lines that intersect at a right angle. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. These two number lines define a flat surface called a plane, and each point on this plane is associated with an ordered pair of real numbers (x, y). The first number is called the x-coordinate, and the second number is called the y-coordinate. The intersection of the two axes is known as the origin, which corresponds to the point (0, 0).
Rectangular Coordinate System

The x- and y-axes break the plane into four regions called quadrants, named using Roman numerals I, II, III, and IV. The ordered pair (x, y) represents the position of points relative to the origin. A few ordered pairs are plotted below:
Quadrants in the Cartesian Coordinate System

This system is often called the Cartesian coordinate system, named after the French mathematician RenĂ© Descartes (1596–1650). Read about him on Wikipedia, he was a very interesting person!


A linear equation has standard form,
ax + by = c
 
where a, b, and c are real numbers and a and b are not both zero. With numbers a linear equation looks like,
6x  3y = 12
Solutions to linear equations are ordered pairs (x, y) where the coordinates, when substituted into the equation, produce a true statement. For example, to show that (3,2) and (−2, −8) are solutions substitute and then simplify:
Here we can see (3, 2) and (−2, −8) are solutions to the equation. From geometry we know that two points determine a line.  So at this point we can plot these points and then draw a line through them.  The line represents all ordered pair solutions and is called its graph.
Graph of A Line

We have graphed our first line!

Tip 1: The scale on the x-axis does not need to match the scale on the y-axis.  Choose a scale that is convenient. In this case, each tick on the y-axis represented 2 units. However, when you do this you need to be careful when plotting the points.

Tip 2: You do not really need graph paper, but it sure does help. Go to Google and search “free printable graph paper.” Or visit OpenAlgebra.com and click on the link to printable graph paper on the right and print some out.

Tip 3: Use a straightedge – a ruler or ATM card. All lines are straight, do not try to freehand them especially when accuracy is important.

Notice that (−3, 2) is not on the line and so these coordinates should not solve the equation.
Now, the question is, “how did we get those points?”  Read [ Graphing Lines by Plotting Points ] and you will see.
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Graphing Lines – The Definitive Guide

Many students do not enjoy graphing and find it very difficult to do. This no-nonsense guide is designed to ease the pain and quickly get you up to speed. You will learn three methods for graphing lines. Each method is laid out in clear and concise way showing each and every step.  It’s easier than you think, I will show you how.  As a bonus, I will also show you how to find equations of lines, parallel or perpendicular, using two easy methods.

Table of Contents: ( COMING SOON: I will post and link as I go.)
Part I:
  1. The Rectangular Coordinate System
  2. Graphing Method 1: Plotting Points 
  3. Graphing Method 2: Using Intercepts
  4. Graphing Method 3: Using the y-intercept and Slope

Bonus:
  1. Finding Equations of Lines – Parallel and Perpendicular
  2. Equations Method 1: Using Slope-Intercept Form
  3. Equations Method 2: Using Point-Slope Form

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Friday, July 26, 2013

Graphing Logarithmic Functions

One way to graph logarithmic functions is to first graph its inverse exponential. Then use your knowledge about the symmetry of inverses to graph the logarithm. Recall that inverses are symmetric about the line y = x. For example,
Step 1: Find some points on the exponential f(x). The more points we plot the better the graph will look.
 
Step 2: Switch the x and y values to obtain points on the inverse.
 
Step 3: Determine the asymptote.
 
In practice, we use a combination of techniques to graph logarithms.  We can use our knowledge of transformations, techniques for finding intercepts, and symmetry to find as many points as we can to make these graphs. General guidelines follow:

1. Graph the vertical asymptote. All logarithmic functions of the form  
have a vertical asymptote at x = h.

2. Find the x- and y-intercepts if they exist. To find x-intercepts set y = f(x) to zero and to find y-intercepts set x = 0.

3. Plot a few more points and graph it.

Graph the following logarithmic functions. State the domain and range.
 
 
In the previous solved problem, make a note of the rigid transformations.  If we start with the basic graph y = log(x) then the first translation is a shift to the left 3 units y = log(x+3).  Next we see a vertical shift up 2 units y = log(x+3)+2 .
 
 
 
 
In the above problem there was a reflection about the x-axis as well as a shift to the left 3 units.
 
 
 
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