How do you graph # f(x)= (2x^2+5x-12)/(x+4)#?

Question

Isaiah Anthony · Accepted Answer

Factor the numerator using the #a*c# method. Cancel common terms in the numerator and denominator. Make a table of #x# and #y# values. Plot the points, and draw a straight line through the points.

Substitute #y# for #f(x)#. #y=(2x^2+5x-12)/(x+4)# Factor the numerator using the #a*c# method. #2x^2+5x-12# #ax^2+bx+c# #a=2;# #b=5;# #c=-12# #a*c=2*-12=-24# Find two numbers that when multiplied equal #-24# and when added equal #5#. The numbers #-3# and #8# fit the criteria. Rewrite #5x# as #-3x# and #8x#. #2x^2-3x+8x-12# Group and factor. #(2x^2-3x)+(8x-12)# = #x(2x-3)+4(2x-3)# = #(x+4)(2x-3)# Rewrite the numerator as #(x+4)(2x-3)#. #y=((x+4)(2x-3))/(x+4)# Cancel #(x+4)#. #y=(cancel(x+4)(2x-3))/cancel(x+4)# = #y=2x-3# Make a table of #x# and #y#. Plot the points, and draw a line through the points. Table of #x# and #y# values. #x=-2;# #y=-7# #x=0;# #y=-3# #x=2;# #y=1# graph{y=2x-3 [-11.3, 11.2, -7.56, 3.69]}

Isaiah Anthony · Answer

undefined To graph the function f(x) = (2x^2 + 5x - 12)/(x + 4), follow these steps:

1. Determine the domain of the function by finding the values of x for which the denominator (x + 4) is equal to zero. In this case, x cannot be -4.

2. Find the x-intercepts by setting the numerator (2x^2 + 5x - 12) equal to zero and solving for x. Factor or use the quadratic formula to find the x-values.

3. Find the y-intercept by substituting x = 0 into the function and solving for y.

4. Determine the vertical asymptotes by finding the values of x for which the function is undefined. In this case, x cannot be -4.

5. Determine the horizontal asymptote by analyzing 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 degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.

6. Plot the x-intercepts, y-intercept, vertical asymptotes, and horizontal asymptote on a coordinate plane.

7. Choose additional x-values and evaluate f(x) to plot more points on the graph.

8. Connect the plotted points smoothly to form the graph of the function.

Remember to label the axes and any important points on the graph.