Question

Y=f(x)/g(x)? 
F(x)=2x^4+7x^3-2x^2+7x-4 
G(x)=3x^4-16x^3-24x^2+64x+48 
H(x)=x^5-3x^4-2x^3+6x^2+x-3 


Find the x-intercept, y-intercept, vertical asymptote, horizontal asymptote, holes and sketch.

Answer 1

The functions are f(x) = 2x⁴  + 7x³ - 2x²  + 7x - 4

g(x) = 3x⁴ - 16x³ - 24x² + 64x + 48

y = f(x)/g(x)

y = (2x⁴  + 7x³ - 2x²  + 7x - 4)/(3x⁴ - 16x³ - 24x² + 64x + 48)

The rational function can be written as,

y = [(2x - 1)(x + 4)(x² + 1)]/[(3x + 2)(x + 2)(x - 2)(x - 6)]

There is no common factors to cancel in the numarator and denominator of the above rational function.

Holes : none.

Find the intercepts

To fin x intercept substitute y = 0 in function.

0 = [(2x - 1)(x + 4)(x² + 1)]/[(3x + 2)(x + 2)(x - 2)(x - 6)]

(2x - 1)(x + 4)(x² + 1) = 0

x = 1/2, x = - 4 and x = -i, i

x intercepts are 0.5 and - 4.

To fin y intercept substitute x = 0 in function.

y = [(2(0) - 1)(0 + 4)((0)² + 1)]/[(3(0) + 2)(0 + 2)(0 - 2)(0 - 6)]

y = [ (- 1) (4) (1)/(2)(2)(-2)(-6)]

y = [ - 4/48]

y = - 1/12(3x + 2)(x + 2)(x - 2)(x - 6)

y intercept is - 0.083.

Vertical asymptote can be found by making denominator = 0.

(3x + 2)(x + 2)(x - 2)(x - 6) = 0

x = -2/3, x = -2, x = 2 and x = 6

Vertical asymptote is at x = 1.

To find horizontal asymptote, first find the degree of the numerator and  the degree of denominator.

Degree of the numerator = 4 and the degree of denominator = 4.

Since the degree of the numerator is equal to the degree of the denominator,horizontal asymptote is the ratio of the leading coefficient of numerator and denominator.

Leading coefficient of numerator =  2, leading coefficient of denominator = 3

y = 2/3 is the horizontal asymptote.

Comments

Contd...

We need some more points to more accurate graph.

Choose random values for x and find the corresponding values for y.

x	y=[(2x-1)(x+4)(x2+1)]/[(3x+2)(x+2)(x-2)(x-6)]	(x, y)
-3	y=[(-6-1)(-3+4)(9+1)]/[(-9+2)(-3+2)(-3-2)(-3-6)]	(-3,-0.22)
-1	y=[(-2-1)(-1+4)(1+1)]/[(-3+2)(-1+2)(-1-2)(-1-6)]	(-1,0.857)
3	y=[(6-1)(3+4)(9+1)]/[(9+2)(3+2)(3-2)(3-6)]	(3, -2.12)
4	y=[(8-1)(4+4)(16+1)]/[(12+2)(4+2)(4-2)(4-6)]	(4,-2.83)

Graph

1) Draw the coordinate plane.

2) Next dash the horizontal and vertical asymptotes

3) Plot the x,y intercept and coordinate pairs found in the table..

4) Connect the plotted points .

When you draw your graph, use smooth curves complete the graph.