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General form

We are going to study functions like these:

L1(x) = 2*x+3

L2(x) = -12*x -50

L3(x) = 0.1*x+11.5

In other words, all of these functions have the form:

L(x) = a*x+b

where a and b are numbers and x is the variable.  And to keep the writing to a minimum, we will drop the "*" and just write:  L(x) = ax+b.

The engineer's trick

When engineers are asked to study a problem, they will usually try to find the extremes first. They will ask questions like:  "what if we make this very large, or make that very small, what would happen?"

We will apply this trick often.  In the formula L(x) = ax+b there are only two numbers:  a and b.

First let's see what happens when b=0.  Then our function is just L(x) = ax.  Let's explore this with Grapher.

L(x) = ax

Make a new window, erase the y= bit and type a=3:

The technical name for a is parameter, but we'll come to that later.  Now click the + button to add a function and type y=ax , select the function and colour it red :

That's quite right: when x=1 the value of ax with a=3 will be 3*x = 3*1 = 3.  If you want to see this precisely, select the menuitem Equation —> Evaluation… to get this panel:

Make sure that the check boxes for Tangent, Perpendicular and Osculating Circle are checked off, but Position is on.  Type 1 into the x field and press enter:

You may have to change the axes numbering by changing the spacing.  To do that, click the x-axis:

(little "balls" appear), then click the inspector and move the spacing scrollbar until you see the numbers you want. Do the same for the y-axis (vertical axis).

Note that you have to close the evaluation panel if you want to do anything else than change the x-field.  Even to zoom in or out you have to close that panel first.  Play a little with the evaluation panel.

Then change the value of a to 0.5:

The red line now slopes less than before.  Try all sorts of values for a, including negative ones!

The Slope

First, you will have noticed that L(x)=ax is indeed a straight line, though that's not so easy to prove and we will not do it here.

Second, you will have noticed that whatever value you give to a, the line L(x)=ax always goes through the origin, the point x=0,y=0 where the x-axis and the y-axis cross.

Indeed, if x=0 then obviously, whatever the value of a, ax will be zero too.

Thirdly, you will have noticed that if a is a big positive number the line slopes up to the right quite steeply, but if a is a negative number it slopes down.  In fact, the only difference a makes is in the slope.

a=0

Doing our engineer's trick again, let's make a=0 and just look at what that does.  Our function now is reduced to L(x)=b.

That's a constant.  Make a new equation by pressing the + button, type b=4.  Grab the line and move it up to just under the equation for a:

change the equation for y to y=ax+b:

Finally make a equal to 0:

We get a horizontal line.  That's quite right:  whatever we choose for x, ax+b will effectively be just b since a is zero, and therefore the value of L(x) will just be b always.

Final study of the function L(x)=ax+b

Now let's make a=0.8 and b=1.9.  We get:

First:

when x=0 we will get that ax+b is just 0x+b or just b.  And indeed we see that the line crosses the vertical axis at the point 1.9.

Second:

Each time we increas x by 1 the value of L(x) will increase by a: if x=4 then L(x)=0.8*4+1.9 and if we now make x=5 then L(x)=0.8*5+1.9 and the difference is obviously just 0.8*1.

So we can make this illustration:

Conclusions:

For any two numbers a and b:

The graph of the function L(x)=ax+b is a line.

The line crosses the y-axis at a point with ordinate b.

If a<0 then the line slopes down.  If a=0 the line is horizontal.  if a>0 then the line slopes up.

If b=0 then the line goes through the origin and the function L(x)=ax is called a proportionality.

If b≠0 then L(x)=ax+b is called a linear relation or just a line.

What about vertical lines?  The engineer's trick would be to say:  what happens when a becomes infinitely large? Then the slope is very steep, in fact it would be vertically up.  But mathematically this is nonsense, because for a vertical line x always has the same value, so it has no meaning to change the x in L(x)=ax+b.

Vertical lines are represented by stating:  x=c where c is a number.

And that is all there is to know about how the functions of the form L(x)=ax+b appear.

But there is more:  intersections of lines.

However, before we go there we will use a much nicer form of the equation of a line: the normalised form.


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