2.2 - Solving Equations Graphically

Intercepts

x-intercept

The x-intercept is the point where the graph of an equation crosses the x-axis. The x-intercept can be found by substituting y=0 into the equation and solving for x. The x-intercept is also called a solution, root, or zero of the equation.

y-intercept

The y-intercept is the point where the graph of an equation crosses the y-axis. The y-intercept can be found by substitution x=0 into the equation and solving for y.

There is no requirement that an equation have either an x-intercept or y-intercept. It is also possible that there may be more than one of each intercept.

Intercept, Root, Zero, Solution?

The following statements are equivalent:

• The point (a,0) is an x-intercept of the graph of y=f(x)
• x=a is a zero or root of the function f
• x=a is a solution of the equation f(x)=0
• (x-a) is a factor of f(x)

Finding roots with the Graphing Calculator

To find the roots of a function using the graphing calculator, it may be necessary to put the equation into standard form f(x)=0. To accomplish this, you may need to move everything over to one side of the equation. It does not matter which side you move the terms to, the solutions will be the same either way. Then, graph the equation y=f(x) on the calculator.

The book teaches you to use the zoom and trace features of your calculator to find the zeros of a function. There is a much easier and more accurate way.

1. Use the [Calc] (2^nd Trace) key and choose the zero (TI83) or root (TI82) option.
2. When it asks for the left or lower bound, use the left arrow key to position the cursor to the left of the zero and press the [Enter] key.
3. When it asks for the right or upper bound, use the right arrow key to position the cursor to the right of the zero and press the [Enter] key.
4. When it asks for a guess, use the left arrow key to move back to the point closest to the x-intercept and press the [Enter] key.
5. The x-coordinate returned is the x-intercept, root, zero, or solution that you're looking for.

The zoom and trace method will work, but it is a whole lot faster, and more accurate, to use the options under [Calc].

Finding Intersections with the Graphing Calculator

To find the intersection of two equations with the graphing calculator, enter the two functions into the calculator as y₁ and y₂ and graph them both. Play around with the viewing window until you find the point of intersection, and then ...

1. Use the [Calc] key and choose the intersect option.
2. It will ask for the first curve. If there are only two plots active, then you can just hit [Enter] as it defaults to the first curve. If you have more than two plots active, then use the up or down arrow keys to select the proper function, and then press enter. It does not matter where on the curve you press enter.
3. It will ask for the second curve. If there are only two plots active, then you can just hit [Enter]. Use the up or down arrow keys if you need to select a different function.
4. It then asks for a guess. Use the left or right arrow keys to move to a position close to the point of intersection and press [Enter].
5. The x-coordinate and y-coordinate of the point of intersection are returned.

It is possible to find the solution to an equation in non-standard form by letting the y₁ plot be the left hand side of the equation, the y₂ plot being the right hand side of the equation, and using the intersect option. However, it is much more convenient and time efficient to rewrite the equation in standard form and use the zero or root option instead.