Unit 7 Rational Functions Topic Transformations of the

Unit 7 – Rational Functions Topic: Transformations of the Rational Parent Function

Rational Parent Function �Graph of the rational parent function is a hyperbola. �Vertical asymptote at x = 0; D: {x | x ≠ 0} �Horizontal asymptote at y = 0; R: {y | y ≠ 0} �Asymptote: boundary line for the graph of the function.

Transforming the Rational Parent Function �General format of a rational function: �Possible transformations (we’ve done this before): ◦ |a| > 1: stretches hyperbola away from origin. ◦ |a| < 1: compresses hyperbola towards origin. ◦ a < 1: reflects graph across x-axis. ◦ h : translates function left or right. �Moves the vertical asymptote. �Vertical asymptote is the line x = h; D: {x | x ≠ h} ◦ k : translates function up or down. �Moves the horizontal asymptote. �Horizontal asymptote is the line y = k: R: {y |y ≠ k}

Vertical Asymptote �Set the Denominator of the function to zero and solve. �x – 4 = 0; x-4+4=0+4; x=4 �The Vertical Asymptote is the vertical line with equation x=4 and the Domain is {x x≠ 4} Note: When graphing choose 3 x-values to the LEFT of the Vertical Asymptote and 3 x-values to the RIGHT.

Horizontal Asymptote �To find the horizontal asymptote if the numerator is a constant then the Horizontal Asymptote is y=0, otherwise set the term with the greatest degree of the numerator over the term with the greatest degree in the denominator and reduce (simplify the terms). The numerator is a constant, 3, therefore the HA is y=0 and the Range is {y y≠ 0}. The numerator term with greatest degree is 2 x and the denominator term with the greatest degree is x set them in a new fraction and simplify, the Range is {y y≠ 2}.

Holes in the Graph �When a factor in numerator cancels with a factor in the denominator �Set the factor that cancelled = 0 and solve. x=0 is the hole on the graph.

Slant Asymptotes Watch this You. Tube video to find out how.

Transforming the Rational Parent Function �Identify the asymptotes, domain & range for the given function, then sketch the graph of the function. n n n V. asymptote: x = – 2 (remember to change the sign for h) H. asymptote: y = 4 D: {x | x ≠ – 2}; R: {y | y ≠ 4} • Plot asymptotes • Since everything shifted left 2 & up 4, the points (1, 1) & (– 1, – 1) from the parent function are now ( – 1, 5) & (– 3, 3). Plot these points. • Sketch the resulting hyperbola through those points.

Transforming the Rational Parent Function �Using the rational parent function as a guide, describe the transformations and graph the function. n n The function will translate 3 units right and 6 units down from the parent function. V. asymptote: x = 3 H. asymptote: y = -6 Plot anchor points and sketch the function.

Asymptote Review � Vertical Asymptotes-set denominator = 0. Horizontal Asymptotes: 1. Constant in numerator-HA is y=0. 2. Degree is equal in numerator and denominator y = reduced answer. 3. The degree of the numerator is one smaller than the degree of the denominator, then y = 0 (since the top will simplify to a constant). 4. Degree in the numerator is one bigger than the degree in the denominator then y = (slant asymptote equation).

Homework Textbook Section 3 -7(pg. 186): 14 -23, 24 -28 even Extra Credit: page 187 30 -40 done in their entirety, all work shown and graphs drawn (x-values 3 to the left and 3 to the right of VA).

Exit Ticket TITLE: Rational Functions 3 -2 -1 Identify 3 things you already knew from the Power. Point, 2 new things you learned, and one question you still have.