Showing posts with label graph. Show all posts
Showing posts with label graph. 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 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!

Tuesday, April 21, 2015

Printable Lined Graph Paper

Free printable 8.5 in. by 11 in. graph paper templates in PDF or Docx formats.

Classic Horizontal Lined [ PDF ] [ Docx ]
Dots every 0.25 inch, used for graphing.

Classic Squared Graph Paper - [ PDF ] [ Docx ]
Squares 0.2 inch, perfect for graphing.

* The word templates were made by inserting a lined image as a watermark.

Creative Commons License
Printable Graph Paper by is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at

Wednesday, October 23, 2013

Logarithms, Exponentials, and Equations

Notes and video examples covering the following topics:

Function Composition:
Inverse Functions:
Exponential Functions and Their Graphs
The Natural Exponential Function
Logarithmic Functions
Graphing Logarithmic Functions

Change of Base Formula
Properties of the Logarithm
Solving Exponential Equations
Solving Logarithmic Equations
Compound Interest Problems

Exponential Growth and Decay