Entry Task The inverse variation xy = 8 relates the constant speed x in mi/h to the time y in hours that it takes to travel 8 miles. Graph this inverse variation. Then use the graph to estimate how many hours it would take to travel 8 miles jogging at a speed of 4.5 mi/h. Possible answer: 1 h 45 min

Entry Task Factor 1) x2 – 9 2) x2 + 7x – 18

RATIONAL FUNCTIONS A rational function is a function of the form: where p and q are polynomials

Learning Target I can… Identify excluded values of rational functions. Graph rational functions.

A rational function is a function whose rule is a quotient of polynomials in which the denominator has a degree of at least 1. In other words, there must be a variable in the denominator. The inverse variations you studied in the previous lesson are a special type of rational function. Rational functions: Not rational functions: For any function involving x and y, an excluded value is any x-value that makes the function value y undefined. For a rational function, an excluded value is any value that makes the denominator equal to 0.

What would the domain of a rational function be? We’d need to make sure the denominator  0 Find the domain. If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values.

The graph of looks like this: If you choose x values close to 0, the graph gets close to the asymptote, but never touches it. Since x  0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0

Finding Asymptotes VERTICAL ASYMPTOTES There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0 So there are vertical asymptotes at x = 4 and x = -1. VERTICAL ASYMPTOTES Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.

HORIZONTAL ASYMPTOTES We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes. 1 < 2 degree of top = 1 If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0. 1 degree of bottom = 2

HORIZONTAL ASYMPTOTES The leading coefficient is the number in front of the highest powered x term. If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at: y = leading coefficient of top leading coefficient of bottom degree of top = 2 1 degree of bottom = 2 horizontal asymptote at:

OBLIQUE ASYMPTOTES If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder. degree of top = 3 degree of bottom = 2 Oblique asymptote at y = x + 5

SUMMARY OF HOW TO FIND ASYMPTOTES Vertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve. To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator. If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0) If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom If the degree of the top > the bottom, oblique asymptote found by long division.

Example 2A: Identifying Asymptotes Identify the asymptotes. Step 1 Write in y = form. Step 2 Identify the asymptotes. vertical: x = –7 horizontal: y = 0

Check It Out! Example 2a Identify the asymptotes. Step 1 Identify the vertical asymptote. x – 5 = 0 Find the excluded value. Set the denominator equal to 0. +5 +5 x = 5 Add 5 to both sides. x = 5 Solve for x. 5 is an excluded value.

Check It Out! Example 2b Identify the asymptotes. Step 1 Identify the vertical asymptote. 4x + 16 = 0 Find the excluded value. Set the denominator equal to 0. –16 –16 4x = –16 Subtract 16 from both sides. x = –4 Solve for x. –4 is an excluded value.

Check It Out! Example 2b Continued Identify the asymptotes. Step 2 Identify the horizontal asymptote. c = 5 y = 5 y = c Vertical asymptote: x = –4; horizontal asymptote: y = 5

Check It Out! Example 2c Identify the asymptotes. Step 1 Identify the vertical asymptote. x + 77 = 0 Find the excluded value. Set the denominator equal to 0. –77 –77 x = –77 Subtract 77 from both sides. x = –77 Solve for x. –77 is an excluded value.

Check It Out! Example 2c Continued Identify the asymptotes. Step 2 Identify the horizontal asymptote. c = –15 y = –15 y = c Vertical asymptote: x = –77; horizontal asymptote: y = –15

Find the domain, points of discontinuity, and x- and y-intercepts of each rational function. Determine whether the discontinuities are removable or nonremovable. Domain - All Reals except 2; x = 2; X-and y- intercepts (-1,0)(-3,0) and (0,3/2) Non Removable Domain - All Reals except -3; x = -3; X-and y- intercepts (4,0)(0,-4) Removable HINT: Points are removable if the discontinuity caused by (x-a) in the denominator is also in the numerator.

Find the vertical asymptotes and holes for the graph of each rational function. Vertical Asymptote at x = 3 Hole at x = -3 Vertical Asymptote at x = 1 Hole at x = 0 No Vertical Asymptote or holes

Find the horizontal asymptote of the graph of each rational function.

Graph the function.   Steps: 1) Check degrees of numerator and denominator 2) Factor the numerator and denominator Asymptotes? Holes? 3) Find the x and y intercepts and plot 4) Plot a few extra pts 5)Sketch

Homework p. 521 #1-12

Entry Task - Thursday Getting Ready Pg. 515

Thursday Homework P. 521 #13-41 odds

Example 2B: Identifying Asymptotes Identify the asymptotes. Step 1 Identify the vertical asymptote. 2x – 3 = 0 Find the excluded value. Set the denominator equal to 0. +3 +3 2x = 3 Add 3 to both sides. Solve for x. Is an excluded value.