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A rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = . Its graph is a.

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Presentation transcript:

BELL-WORK

Due tomorrow: PW 11-1 # 1-12(even) PW 11-2 # 1-5,14-18 HW 4.4(f) Due tomorrow: PW 11-1 # 1-12(even) PW 11-2 # 1-5,14-18

13; (5, ½) 5; (4, 2½) 17; (3½, 5) 10; (-1, 1) 6½; (1¾, 2) 12½; (0, 1¼) HW 4.4(e) Solutions 13; (5, ½) 5; (4, 2½) 17; (3½, 5) 10; (-1, 1) 6½; (1¾, 2) 12½; (0, 1¼)

Guiding question: How are rational functions simplified?

Rational Functions A rational function has a polynomial of at least degree one in the denominator. Examples: y = 1 x y = 1 . x2+2 y = 3x + 9 x + 3

Simplifying Rational Functions To simplify rational functions: factor the numerator, then factor the denominator, and then carry out operations as usual. Simplify: 3x + 9 x + 3 4x + 20 . x2 – 9x + 20 3x – 27 81 – x2

Simplifying Rational Functions TB pg 654 Got it 3a, 3b (Simplify only)

Multiplying Rational Functions 7 • 8 = y y2 56 y3 x • x – 2 = x + 5 x – 6 x(x – 2) . (x + 5)(x – 6) *Always leave your answer in factored form!

Multiplying Rational Functions 3x + 1 • 8x = 4 9x2 – 1 3x + 1 • 8x . 4 (3x – 1)(3x + 1) = 8x . 4(3x – 1) = 2x . 3x – 1

Multiplying Rational Functions 5x + 1 • (x2 + 7x + 12) = 3x + 12 = (5x + 1) •(x + 3)(x + 4) 3(x + 4) = (5x + 1)(x + 3) 3

Dividing Rational Functions a2 + 7a + 10 ÷ a + 5 = a – 6 a2 – 36 = (a + 5)(a + 2) ÷ a + 5 . a – 6 (a – 6)(a + 6) = (a + 5)(a + 2) × (a – 6)(a + 6) . a – 6 a + 5 = (a + 2)(a + 6)

Dividing Rational Functions x2 + 13x + 40 ÷ x + 8 = x – 7 x2 – 49 = (x + 5)(x + 8) ÷ x + 8 . x – 7 (x – 7)(x + 7) = (x + 5)(x + 8) × (x – 7)(x + 7). x – 7 x + 8 = (x + 5)(x + 7)

Graphing Rational Functions On the TI graph y = 1 x Copy a sketch of the graph to your page

Graphing Rational Functions What is the domain of the function? All values except x = 0 Notice that the graph gets close to x = 0 but the graph never touches x = 0. An asymptote is a line that the graph gets closer and closer to but never touches or crosses. y = 1 has a vertical asymptote at x = 0. x

Graphing Rational Functions What is range of the function? All values except y = 0 Notice that the graph gets close to y = 0 but the graph never touches y = 0. y = 1 has a horizontal asymptote at y = 0. x Note: the excluded values are the asymptotes!

Graphing Rational Functions How will the graph of 1 look? x + 2 The graph of y = 1 represents a horizontal x+c translation of y = 1 x If c > 0 it is a left-ward translation. If c < 0 it is a right-ward translation.

Graphing Rational Functions What is the domain of 1 ? x + 2

Graphing Rational Functions What is the domain of 1 ? x + 2 x - 2

Graphing Rational Functions Therefore for y = 1 there is a vertical asymptote x + 2 at x = -2 What is the range of y = 1 ? All values except y = 0 Therefore there is a horizontal asymptote at y = 0.

Graphing Rational Functions Without the TI sketch the graph of y = 1 . x – 5 What is the domain? All values except x = 5 Therefore there is a vertical asymptote at x = 5 What is the range? All values except y = 0 Therefore there is a horizontal asymptote at y = 0.

Graphing Rational Functions How will the graph of y = 1 + 4 look? x The graph of y = 1 + c represents a vertical translation of y = 1 If c > 0 it is an up-ward translation. If c < 0 it is a down-ward translation

Graphing Rational Functions What is the domain of y = 1 + 4 ? x All values except x = 0 Therefore there is a vertical asymptote at x = 0 What is the range? All values except y = 4 Therefore there is a horizontal asymptote at y = 4.

Graphing Rational Functions Without the TI sketch the graph of y = 1 – 3 ? x + 4 What is the domain of the graph? All values except x = -4 Therefore there is a vertical asymptote at x = -4 What is the range? All values except y = -3 Therefore there is a horizontal asymptote at y = -3.

Who wants to answer the Guiding question? How are rational functions simplified?