# find the x-intercept ,y-intercept, asymptotes of these rational functions?

1) (4x^2+72x+320)/(x^2+9x-10)

2) (20x^2+20x-40)/(x^2+x)
3) (6x^2)/(x^2-2x-63)

Please make any corrections and help with the missing ones.

1) The rational function

Simplify the rational function.

Factor the numarator 4x2 + 72x + 320

= x2 + 18x + 80

= x2 + 10x + 8x + 80

= x(x + 10) + 8(x + 10)

= (x + 10)(x + 8)

Factor the denominator x2 + 9x - 10

= x2 + 10x - x - 10

= x(x + 10) - 1(x + 10)

= (x + 10)(x - 1)

To find y intercept substitute x = 0 in .

y intercept is -8.

To find x intercept substitute y = 0 in .

x intercept is -8.

Vertical asymptote can be found by making denominator = 0.

x - 1 = 0

Vertical asymptote is x = 1.

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

Degree of the numarator =1 and the degree of denominator = 1.

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

Leading coefficient of numarator =   1, leading coefficient of denominator = 1

y  = 1 is the horizontal asymptote.

2) The rational function

To find y intercept substitute x = 0 in

In this case there is no y intercept.

x intercept can be found by making numarator = 0.

20x2 + 20x - 40 = 0

x2 + x - 2 = 0

x2 + 2x - x - 2 = 0

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

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

x + 2 = 0 and x - 1 = 0

x = - 2 and x = 1

x intercepts are -2 and 1.

Vertical asymptote can be found by making denominator = 0.

x2 + x = 0

x(x + 1) = 0

x = 0 and x = - 1

Vertical asymptotes are at x = 0 and x = - 1

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

Degree of the numarator =2 and the degree of denominator = 2.

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

Leading coefficient of numarator =  20, leading coefficient of denominator = 1

y = 20 is the horizontal asymptote.

3) The rational function

To find y intercept substitute x = 0 in

y intercept is 0.

x intercept can be found by making numarator = 0.

6x2 = 0

x = 0

x intercept is 0.

Vertical asymptote can be found by making denominator = 0.

x2 - 2x - 63 = 0

x2 - 9x + 7x - 63 = 0

x(x - 9) + 7(x - 9) = 0

(x - 9)(x + 7) = 0

x = 9 and x = - 7

Vertical asymptotes are at x = 9 and x = - 7.

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

Degree of the numarator = 2 and the degree of denominator = 2.

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

Leading coefficient of numarator =  6, leading coefficient of denominator = 1

y = 6 is the horizontal asymptote.