Introduction to y =
mx + c, about 'm' and 'c' and graphing.

This topic is part of the
TCS FREE high school mathematics 'Howto Library'.
It introduces the straight line function, y = mx + c, discusses 'm' and 'c', and
shows you how to draw graphs of some different forms of this function.
(See the index page
for a list of all available topics in the library.) To
make best use of this topic, you need to download the Maths Helper Plus
software. Click
here for instructions. 
Theory:
Any function: y = f(x) that graphs as a straight line has an equation of this
form:
y = mx + c
NOTE: This does not include vertical
lines, because they are not functions of 'x': y = f(x). A function of 'x' cannot
have more than one point in the same vertical line.
In the straight line equation: y = mx + c:
'x' and 'y' are the coordinates of the points that satisfy the
function and so lie on the straight line graph.
'm' is the gradient of the straight line graph, and
'c' is the 'y intercept' of the straight line graph.
About 'gradient'...
'Gradient' is a number that represents the steepness of a
straight line. A horizontal line has gradient zero. A 45º line has gradient
1, and a vertical line has infinite gradient.
This diagram shows some different lines and their gradients (the 'm'
values):
The sign of the gradient is important. Positive gradients
(like those in the diagram above) mean that the line is sloping uphill
as you go left to right. If a line slopes downhill going left to right,
then it has a negative gradient, as shown below:
Gradient is an exact quantity. This is how it is defined mathematically...
1. Start with any two points on a straight line, (x_{1},y_{1})
and (x,y):
2. Starting at (x_{1},y_{1}) on the left, then moving to the right along
the line to (x,y), we measure the change in 'x', and
the change in 'y'.
The change in 'x' = x  x_{1}
while the change in 'y' = y  y_{1}
Gradient 'm' is defined as follows:
The Greek capital letter 'Delta' : D
is often used in mathematics to mean a change in some quantity. So the
gradient definition can be written like this:
To measure the change in 'x' and 'y' between the two points in the graph
above, we draw a right triangle beneath the graph. The length of the
vertical leg is: (yy_{1}), and the
horizontal leg has length (xx_{1})...
So the gradient of this line is:
About 'y' intercept...
The 'y' intercept is the 'y' coordinate of where the straight line graph
cuts the 'y' axis:
The symbol 'c' is used to represent the 'y' intercept. So in the example
above, c = 2.
Note: at the 'y' intercept, the 'x'
coordinate is zero.
So the equation of the example straight line graph: y = mx + c is:
y = 0.5x + 2
Method:
Maths Helper Plus can find the
equation of a straight line given two points. It will calculate the gradient and
y intercept showing the working steps. It also displays a labelled diagram of
the situation.
Step 1 Download
the free support file...
We
have created a Maths Helper Plus document containing the completed example from
this topic. You can use this to practice the steps described below, and as a
starting point for solving your own problems.
File
name: 'Straight line given two points.mhp' File size: 10kb
Click here
to download the file.
If
you choose 'Open this file from its current location', then Maths Helper Plus should
open the document immediately. If not, try the other option: 'Save this file to disk', then
run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the
saved file and open it. If you do not yet have Maths Helper Plus installed on your
computer, click here
for instructions.

Step 2 Display
the parameters box
Press the F5 key to display the
parameters box:
The coordinates of two (x,y)
points are entered into the four edit boxes 'A', 'D', 'B' and 'X', as follows:
x_{1} =
A, y_{1} = D, x = B, and y = X.
To
find the equation of the straight line through two points, as well as calculate
the gradient 'm' and y intercept '(0,c)', enter the coordinates of two points
into the edit boxes.
To
enter the coordinates, click on the 'A' edit box with the mouse, then type the x_{1}
coordinate. Now click on the 'D' edit box and type the y_{1}
coordinate. Continue until the four coordinates are entered, then click the
'Update' button.
Step 3 Adjust
the scale of the labelled diagram
If the points lie off the graphing
area of the screen, the scale needs to be reduced. In this case, briefly press
the F10 key enough times until the two plotted points are seen.
You can make the diagram bigger by
holding down 'Ctrl' while you press F10.
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