How do you graph #f(x)=(x-1)/x^2# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Isaiah Anthony · Accepted Answer

See answer below

Given: #f(x) = (x-1)/x^2# This type of equation is called a rational (fraction) function. It has the form: #f(x) = (N(x))/(D(x)) = (a_nx^n + ...)/(b_m x^m + ...)#, where #N(x))# is the numerator and #D(x)# is the denominator, #n# = the degree of #N(x)# and #m# = the degree of #(D(x))# and #a_n# is the leading coefficient of the #N(x)# and #b_m# is the leading coefficient of the #D(x)# Step 1 factor : The given function is already factored. Step 2, cancel any factors that are both in #(N(x))# and #D(x))# (determines holes): The given function has no holes #" "=> " no factors that cancel"# Step 3, find vertical asymptotes: #D(x) = 0# vertical asymptote at #x = 0# Step 4, find horizontal asymptotes: Compare the degrees: If #n < m# the horizontal asymptote is #y = 0# If #n = m# the horizontal asymptote is #y = a_n/b_m# If #n > m# there are no horizontal asymptotes In the given equation: #n = 1; m = 2 " "=> y = 0# horizontal asymptote is #y = 0# Step 5, find x-intercept(s) : #N(x) = 0# #x - 1 = 0; " "=> x"-intercept" (1, 0)# Step 5, find y-intercept(s): #x = 0# #f(0) = 0-1/0^2 = "undefined"# no #y#-intercept. graph{(x-1)/x^2 [-10, 10, -5, 5]}

Ethan Adkins · Answer

undefined To graph the function f(x) = (x-1)/x^2, we can analyze its key features:

1. Holes: The function has a hole at x = 1, as the denominator becomes zero at this point. To find the y-coordinate of the hole, substitute x = 1 into the function.

2. Vertical Asymptotes: The function has a vertical asymptote at x = 0, as the denominator approaches zero as x approaches 0. To determine the behavior of the function as x approaches positive or negative infinity, analyze the degrees of the numerator and denominator.

3. Horizontal Asymptotes: To find the horizontal asymptote(s), compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

4. x-intercepts: To find the x-intercept(s), set f(x) = 0 and solve the resulting equation.

5. y-intercept: To find the y-intercept, substitute x = 0 into the function.

By analyzing these features, you can accurately graph the function f(x) = (x-1)/x^2.