The Denominator of a Rational Function

MCR3U Date ______Gizmo Exploration: GRAPHS OF RATIONAL FUNCTIONS

THE DENOMINATOR OF A RATIONAL FUNCTION px In the upper-left hand corner of the Gizmotm, you will see a rational function in the form y  . The qx roots of the denominator affect the vertical asymptotes in the graph of the function. An asymptote is an imaginary line that the graph of a function approaches but does not reach. Asymptotes are usually shown as dotted lines. The root of a linear factor is the number that makes the factor equal zero. For example, the root of the factor (x − 2) is 2, and the root of (x + 5) is −5. The polynomial made by multiplying those two factors, ƒ(x) = (x − 2)(x + 5), has two roots, 2 and −5, because both numbers make ƒ(x) = 0.

1. Click - next to the numerator of the fraction to remove all the factors of the numerator. Set the denominator to have one factor. Then, set the root in the denominator to 0 by selecting the factor in the denominator and setting the root slider to 0. (To quickly set the slider to a specific number, type the number into the field to the right of the slider, and then press Enter.) This will make the function 1 y  . Turn on Show asymptotes. x a. What are the equations of the asymptotes in the graph of this function? ______b. Using the slider, vary the root of the denominator. How does the root in the denominator relate to the equations of the asymptotes? Explain why.

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c. Turn on Show domain and vary the root of the denominator. How is the domain of the function related to the root?

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2. Increase the number of factors in the denominator of the function. Vary their roots and observe the number of asymptotes on the graph. a. How does the number of roots in the denominator relate to the number of vertical asymptotes?

______b. What happens if there is more than one factor in the denominator with the same root?

______MCR3U Date ______Gizmo Exploration: GRAPHS OF RATIONAL FUNCTIONS

3. Make a rational function with one factor in the numerator and two factors in the denominator. Vary the factor of the numerator and observe how the graph changes. a. What happens when a factor in the numerator is the same as a factor in the denominator? Explain why this happens.

______b. How does this affect your answer about the relationship between the number of roots in the denominator the number of vertical asymptotes in the graph of the function?

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x xx 1 4. How are the graphs of the functions y  and y  different? Explain why. x  2 x 1x  2

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5. For each of the following, write down a function that has the given properties. Then, check your answers by graphing your functions in the Gizmo and sketch the graphs below (draw asymptotes using dotted lines). a. vertical asymptote at x = −2; hole at x = 3 b. vertical asymptotes at x = 4 and x = 0; no holes c. vertical asymptote at x = −1; two factors in the denominator; no holes MCR3U Date ______Gizmo Exploration: GRAPHS OF RATIONAL FUNCTIONS

THE NUMERATOR OF A RATIONAL FUNCTION The roots of the numerator of a rational function are related to the x-intercepts in the graph of the function.

x 1 1. Graph the rational function y  in the Gizmo and to the right. x a. What is the root of the numerator of the function?

______x 1 b. What happens to the graph of y  at that x-value? x

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x  5x 2. Graph y  in the Gizmo. Sketch to the right. x 1 a. How many roots does the numerator of this function have? What are they?

______x  5x b. At which points does the graph of y  cross the x 1 x -axis? What observation can you make about the graph of a rational function and the roots of the numerator of the function?

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x  3x  3 3. Graph y  in the Gizmo. Sketch it to the right. x  3 a. Why isn't x = 3 an x-intercept on this graph?

______b. In general, when will a root of the numerator not be an x- intercept on the graph?

______xx  4x  3 c. What are the x-intercepts of y  ? Check your answer in the Gizmo. x  3

______MCR3U Date ______Gizmo Exploration: GRAPHS OF RATIONAL FUNCTIONS

ASSESSMENT: Cut out the puzzle pieces on the next page. Arrange the pieces on the grid below so that the function properties are adjacent to the appropriate graph and equation.

Equation GRAPH Domain GRAPH Equation

Hole Hole

GRAPH GRAPH

Equation Equation Equation MCR3U Date ______Gizmo Exploration: GRAPHS OF RATIONAL FUNCTIONS Cut out the puzzle pieces on this page. Arrange the pieces on the previous page so that the function properties are adjacent to the appropriate graph and equation. (Note: one of the graphs is already placed for you. Put this piece aside. It will not used.)

Vertical Vertical Horizontal X-intercept X-intercept Asymptote Asymptote Asymptote at x = 2 at x = 4 at x = -2 at x = 3 at y = 1

Domain Domain Hole Hole Hole x   | x  2,1,3 x | x  2,3 at x = -2 at x = -1 at x = 3

x  2x 1 xx  2x  4 x  2x 1x  4 x  3x  4 x  2x  3 y  y  y  y  y  x  2x  3x 1 x  3x  2 x 1x  3x  2 x  2x  3 x  3x  2 MCR3U Date ______Gizmo Exploration: GRAPHS OF RATIONAL FUNCTIONS