Definition: <ul><li>A rational function is a function that can be written in the form of a polynomial divided by a polynomial, </li></ul>
Examples: Because rational functions are expressed in the form of a fraction, the denominator of a rational function cannot be zero, since division by zero is not defined.
Domain: <ul><li>The domain of a rational function is the set of all real numbers, except the x-values that make the denominator zero. </li></ul><ul><li>Example: </li></ul><ul><li>The domain is the set of all real numbers except 0, 2, & -5. </li></ul>
Graphing Rational Functions Step 1: Identify zeros (x-intercepts) of the function. The zeros of this function are found by setting y to 0 and solving for x: We see that when we do this, we find that the numerator determines our zeros. In this case, our zero is found by solving x −1=0, so x = 1 is our zero. Step 2: Identify our vertical asymptotes. We know that this function is undefined when the denominator is 0. So, let us find for which x-values the denominator is equal to zero. Our graph will contain vertical asymptotes at these x-values. We want to solve the equation 2 x 2 +7x+3=0.
So, our graph contains two vertical asymptotes, which represent values of x that our function does not contain in its domain. Our graph will tend towards these x-values, but will never pass through them.
<ul><li>Step 3: Identify our horizontal asymptote. </li></ul><ul><li>By definition, a horizontal asymptote is a y-value that a graph tends toward as the x values approach their extremes (∞ and - ∞) . (Note that this doesn’t mean a graph can’t cross its horizontal asymptote for relatively small x-values.) </li></ul><ul><li>We identify our horizontal asymptote by considering what y-value our graph approaches as x -> ∞ and x -> −∞ . We will do this by dividing every term in the function by the highest degree term in the denominator : </li></ul>Notice that when x -> ∞ , each term that contains an x in the denominator after our last step approaches 0, since we’re dividing constants by more and more. So, as x -> ∞ , . Similarly, as x -> −∞ , y -> 0 . So, our horizontal asymptote is y = 0 . Notice that the degree of the numerator is less than that of the denominator. If this is the case, the horizontal asymptote is y = 0 .
<ul><li>Step 4: Check whether graph crosses horizontal asymptote. </li></ul><ul><li>Since we can cross horizontal asymptotes for relatively small values of x, we will check whether we do so in this example. To cross our horizontal asymptote would mean that y = 0 for some value(s) of x. So, we want to solve the equation . </li></ul><ul><li>We found in Step 1 that x = 1 is the solution to this equation, so our graph does cross our horizontal asymptote when x = 1. </li></ul>Step 5: Find y-intercept. We find our y-intercept by setting x to be 0 and solving for y. So, since our y-intercept is .
<ul><li>Step 6: Graph lies above or below x-axis. </li></ul><ul><li>Our graph is split up into three sections, since we have two vertical asymptotes. One </li></ul><ul><li>section is all x-values less than -3, the next section is x-values between -3 and , and the third section is all x-values greater than . The third section has two sections in itself, since we have a zero in that interval. We want to determine where our graph lies in relation to the x-axis in each of these four sections. We will pick a value of x in each of these sections to determine the sign of the function in each of these sections: </li></ul>Section Test Point Value of function Above or below the x-axis x < -3 x = -4 y < 0 Below -3 < x < x = -1 y = 1 Above < x < 1 x = 0 y = < 0 Below x > 1 x = 2 y = 0.04 Above
<ul><li>Step 7: Graphing our function. </li></ul><ul><li>We put all of the elements we just found in the previous 6 steps together and obtain the following graph: </li></ul>
Let’s look at another one… <ul><li>Graph the rational function: </li></ul><ul><li>Identify the horizontal asymptote </li></ul><ul><ul><li>Since the numerator and the denominator are both the </li></ul></ul><ul><ul><li>same degree (2), you find the horizontal asymptote by </li></ul></ul><ul><ul><li>dividing the leading coefficients . </li></ul></ul><ul><li>y= 1/1 = 1 </li></ul>
Step 2 <ul><li>Let y = 0 to find the possible x intercepts </li></ul>
Step 3 <ul><li>To find the vertical asymptotes, set the denominator equal to </li></ul><ul><li>zero and solve </li></ul><ul><li>Notice that the -2 showed up as a asymptote AND an </li></ul><ul><li>intercept. That means that something is going on around -2. </li></ul><ul><li>Look at this…. </li></ul>
Step 3b <ul><li>If you factor the numerator and the denominator, you </li></ul><ul><li>are able to reduce… </li></ul>When you can reduce, this creates a hole. Think of asymptotes like walls and holes like puddles…you can’t walk through a walk, but you can jump over a puddle. When you see duplicate values, that means there is a hole and not really an asymptote. You can even look for holes first!
Step 4 <ul><li>Find the y intercept by letting x = 0 </li></ul>
Start putting it together When we put the asymptotes, intercepts, and hole on the graph it’s easy to see where the graph goes…
Draw in the graph… <ul><li>**Rational f(x)’s “flip-flop” over the asymptotes** </li></ul>
Using the TI <ul><li>You can use the calculator to graph these, just make sure </li></ul><ul><li>you enter the function correctly! The numerator and the </li></ul><ul><li>denominator need there own parenthesis! </li></ul><ul><li>You will need to enter: (x^2 -4) / (x^2 – x -6) </li></ul><ul><li>You will be able to see the vertical asymptotes ONLY! </li></ul><ul><li>They will appear as a line on the graph. </li></ul><ul><li>The rest is up to you </li></ul>
Practice Problem: <ul><li>Graph the function . Find all asymptotes and x- & y-intercepts. </li></ul>x-intercept: x = 2 y-intercept: y = Vertical asymptotes: x = -3 & x = 4 Horizontal asymptote: y = 0
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