Graphs of Rational Functions

Página 1

y =(ax + b)/(cx + d)

To find asymptotes;

Horizontal; as x -> ∞

Vertical; as y -> ∞

Example 1;

y = (4x - 8)/(x + 3)

x -> ∞

y = (4 - 8/x)/(1 + 3/x)

y = (4 - 0)/(1 - 0)

y = 4

y = (4x - 8)/(x + 3)

y -> ∞

x = -3

Example 2;

y = ((x - 3)(2x - 5))/((x + 1)(x + 2))

y = ((1 - 3/x)(2 - 5/x))/((1 + 1/x)(1 + 2/x))

x -> ∞, 3/x, 5/x,1/x,2/x --> y=2

(x + 1)(x + 2) = 0

x = -1, -2

x = 0, y =7.5

y = 0, (x - 3)(2x - 5) = 0 and x = 3, 2.5

[0 , 7.5] [2.5 , 0] [3 , 0]

2 = ((x - 3)(2x - 5))/((x + 1)(x + 2))

2(x^2 + 3x + 2) = 2x^2 -11x + 15

6x + 4 = -11x +15

x = 11/17

[0.647 , 2]

Página 2

Irreducible quadratic in the denominator;

y = (x^2 + ax + b)/(x^2 +cx + d)

Example 1;

y = (x^2 + 2x - 3)/(x^2 + 2x + 6)

Horizontal Asymptote;

y = (1 + 2/x + 3/x^2)/(1 +2/x + 6/x^2)

x ---> ∞, y --> 1 y = 1; Horizontal

x^2 + 2x + 6, b^2 - 4ac = 4 - 24 = -20

Therefore no vertical asymptote

y = 0, x^2 + 2x - 3 = 0

(x + 3)(x - 1) = 0

x = -3, 1 ---------> [-3 , 0] [1 , 0]

x = 0, y = -0.5. [0 , -0.5]

1 = (x^2 + 2x - 3)/(x^2 + 2x + 6)

x^2 + 2x - 3 = x^2 + 2x + 3 =/= Impossible solution.

Never crosses horizontal

Página 3

Stationary Points;

y = (x^2 + 2x - 3)/(x^2 + 2x + 6)

k = (x^2 + 2x - 3)/(x^2 + 2x + 6)

kx^2 + 2kx + 6k = x^2 + 2x - 3

(k-1)x^2 + (2k - 2)x + 6k + 3 = 0

(2k - 2)^2 - 4(k - 1)(6k + 3) = 0

4k^2 - 8k + 4 -4(6k^2 - 3k - 3) = 0

-20k^2 + k +16 = 0

5k^2 - k - 4 = 0

(5k^2 + 4)(k - 1) = 0

k = -0.8, k = 1

Minimum = -0.8

k = -0.8, (-9/5)x^2 - (18/5)x - (9/5) = 0

x^2 + 2x + 1 = 0

(x + 1)^2 = 0

x = -1

Minimum Coordinates; [-1 , -0.8]

Página 4

Inequalities;

(ax + b)/(cx + d) > k

Method 1;

(ax + b)/(cx + d)>k

Cross multiply by (cx + d)^2

((ax + b)(cx + d)^2)/((cx + 4)(cx + d)^2)

==> ((ax + b)(cx + d)(cx + d))/((cx + d)(cx + d)(cx + d))

==> ((ax + b)(c^2x^2 + 2dcx + d^2))/((c^2x^2 + 2dcx + d^2)(cx + d))

==> ((ac^2x^3 + 2adcx^2 + ad^2x + bc^2x^2 + 2bdcx + bd^2))

==> ((c^3x^3 + 2dc^2x^2 + cd^2x + dc^2x^2 + 2d^2cx + d^3)

===> (((ac^2)x^3 + (2adc)x^2 + (ad^2)x + (bc^2)x^2 + (2bdc)x + bd^2))

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===> (((c^3)x^3 + (2dc^2)x^2 + (cd^2)x + (dc^2)x^2 + (2d^2)x + d^3)

===> (((ac^2)x^3 + (2adc + bc^2)x^2 + (ad^2 + 2bd)x + bd^2))

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===> (((c^3)x^3 + (3dc^2)x^2 + (3cd^2)x + bd^2))

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Method 2;

Sketch y = (ax + b)/(cx + d)

Solve k = (ax + b)/(cx + d)

Compare results