# Sketch each of the following curves. Show all work. State all holes, asymptotes, roots, intercepts, and end behaviours f(x)=2x^3-13x^2+13x+10/x-1and part b) f(x) = x^2 + 4x + 3/ x-2

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the roots of function f(x) =  such that:

2x^3-13x^2+13x+10/(x-1) = 0

You need to select one of divisors of 10 to veri:fy if it may be a root for f(x) such that:

x = 2

16 - 52 + 26 + 10 = 0

0 = 0

Notice that x = 2 is a root for polynomial, hence you may write the factored form of f(x) such that:

f(x) = (x-2)(ax^2(x-1) + bx(x-1) + c(x-1))

f(x) = (x-2)(x-5.2)(x^2 - 2/10x + 4/10)

You should solve the equation x^2 - 2/10x + 4/10 = 0 such that:

10x^2 - 2x + 4 = 0

x_(3,4) =  (2+-sqrt(4 - 160))/20 => x_(3,4) = (2+-i*sqrt156)/2

x_(3,4) = 1 +-6.24 i

Hence, the graph of f(x) intercepts x axis at x = 2 and x = 5.2 and it has a vertical asymptote at x = 1.

You need to sketch the graph of function f(x) = x^2 + 4x + 3/ (x-2) such that:

x^2 + 4x + 3/( x-2) = 0

x^3 - 2x^2 + 4x^2 - 8x + 3 = 0

x^3 + 2x^2 - 8x + 3 = 0

You need to find the extreme points of function, hence you should evaluate the derivative of function such that:

f'(x) = 3x^2 + 4x - 8

You need to solve the equation f'(x) = 0 such that:

3x^2 + 4x - 8 = 0

x_(1,2) = (-4+-sqrt(16 + 96))/6

x_(1,2) = (-4+-10.58)/6

x_1 = 1.09 ; x_2 = -2.43

Hence, the function reaches its maximum at x = -2.43 and it reaches its minimum at x = 1.09.

The function has a vertical asymptote at x = 2.
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