Algebra/Intercepts

Intercepts
To find where the equation of a line crosses the X or Y axis, you don't need much information. In this section we will look at how to find the where the line crosses the axes using the standard form for the linear equation. After we look at how slope works we will see we can convert between the various types of linear equations into the standard form.

X and Y axis intercepts
An axis intercept point is a point where the graph of a function, relation, or equation intersects the X or Y axes. This section is about finding out how a particular set of functions: linear functions cross the axes.

We know that the domains of most lines are infinite because they are defined at every value of X. The exception is lines that are defined as $$X=c$$ where c is a number we choose when we write the function. By definition this line is only defined for one value of X. Since the domain maps onto more than one value for the range this is actually a relationship and not a function. We've seen the graph of this relationships is a vertical line that passes through the point (c,Y). In the picture below we show that when c=0 then our line is the same as the Y axis. When $$ c \ne 0 $$ (as in the drawing where X = 3) then the line can never intercept the Y axis.

Lines with the equation X=C intersect the X axis once and the Y axis 0 or infinite times.

We can also restrict the range of a function by simply writing $$Y=c$$ where c is again any number we choose. The graph of this line is a horizontal line that passes through the point (X,c). When Y=o this line is the same as the X axis. when $$c \ne 0 $$ (as in the drawing where Y = 3)then the line can never intercept the X axis.

Lines with the equation Y=C intersect the Y axis once and the X axis 0 or infinite times.

When looking at Cartesian graphs and linear equations we run into a mathematical axiom: "Two points determine a line.". We will see how this axiom affects the slope-intercept definition of a line $$y= f(x) = m x + b$$ in the next section. When two lines intersect they intersect at a point. If a line is not horizontal or perpendicular it will have to intersect the X and Y axes once, but only once.

In this book we are going to accept the statement "At most one line can be drawn through any point not on a given line parallel to the given line in a plane." There is a branch of mathematics called "non-euclidean geometry" that was founded a little more than 160 years ago. Even if you are not interested in mathematics it is worth looking at this Wikipedia article on geometry to get a feel for how formalizing geometry with algebraic methods and then moving beyond them has changed civilization. If you continue in a career requiring advanced mathematics such as Engineering or Physics you might want to follow your interests to see the effect of non-euclidean geometry in your career.

We've seen that for the equation Y=mX + b the Y intercept will always be at b because that is where X=0.

Using Algebra we can subtract b from both sides: Y - b = mX

and multiply by $$\frac {1}{m}$$

$$ \frac {Y}{m} - \frac {b}{m} = X$$

we can see that the X intercept is going to be $$-\frac{b}{m}$$.

An axis intercept may simply refer to the number value on the axis where the intersection occurs. For brevity we may say the line has an X intercept of 1 and a Y intercept of 2. After graphing just a few lines you will be able to tell this line points down and runs through quadrants II, I, and IV. With a little more practice you will be able to know that the equation for the line is Y=-2x + 2. We will see that by specifying the two points we are actually implying the slope of the line. There is an exception to this rule. If we say a line crosses the axes at 0 we know that the line will pass through 2 quadrants instead of 3, but we won't know which quadrants or how steep the line is. When we look at slope in the next section we will see why the equations above specify a point and a slope.

When you are trying to graph a linear equation finding the axes intercepts is often the easiest way to go about doing it. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y.  For most examples the intercepts are different points, and a line can be drawn through the two intercepts. If both intercepts are (0,0), then another point must be determined to graph the line. If the equations is in the form x = c or y = c, the horizontal or vertical lines are very simple to plot.

Example
Original equation
 * $$Y = 5 \times x + 2$$


 * $$Y = 5 \times 0 + 2$$

Substitute zero for x


 * $$Y = 2$$

Solution

Therefore, the Y-Intercept of Y = 5x + 2 is 2.

This works for any form of equation.