How do you graph
f(x)= ((x+5)(x-4)^(2))/((x-2)(x^(4))
First of all some critical points.
From the single factor of (x+5) at the top, we can say it crosses at (-5,0)
from the double factor of (x-4)^2, the graph will "touch" the x-axis and then reverse direction.
From the x^4 at the bottom, the y-axis will be a vertical asymptote,
From the (x-2)at the bottom, there will be a vertical asymptote at x=2
The highest power at the top will be +x^3 and the highest power at the bottom will be +x^5, so as x approaches infinity in either the positive or negatives, it will approach zero and the graph will approach the x-axis from the top.
I tried some positive and negative values of x close to zero and the function value was negative. Also a value of x = 1.9 gave me a negative value. So the "loop between 0 and 2 lies below the x-axis, suggesting that there is a maximum value for that part of the graph, but still below the x-axis
Also after (4,0), the graph will rise ever so slightly for a maximum just to the right of (4,0) and then approach the x-axis.
The same thing will happen on the left at (-5,). The graph will come up from its y-axis asymptote, cross at (-5,0), rise just ever so slightly and then drop down to approach the x-axis
If you have a graphing calculator, you can zoom in on these critical areas, but on a large scale the small changes near (4,0) and (-5,0) will be hardly noticeable.
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To graph a function, you need to follow a step-by-step process. In the case of the function f(x) = ((x+5)(x-4)^(2))/((x-2)(x^(4))), here are the steps you can take to graph it:
Step 1: Determine the domain of the function.
The domain is the set of all possible x-values for which the function is defined. In this case, the function is defined for all real numbers except x = 2 and x = 0 (which would make the denominator zero due to the factors (x-2) and x^4 respectively).
Step 2: Find the y-intercept.
The y-intercept is the point where the graph intersects the y-axis. To find it, substitute x = 0 into the function: f(0) = ((0+5)(0-4)^(2))/((0-2)(0^(4))). Simplifying this, we get f(0) = (5*(-4)^2)/((0-2)(0^4)) = (5*16)/((0-2)(0)) = -40/(-2*0) = undefined. Since the y-intercept is undefined, there is no y-intercept for this function.
Step 3: Determine the x-intercepts.
The x-intercepts are the points where the graph intersects the x-axis. To find them, set f(x) = 0 and solve for x. In this case, we have ((x+5)(x-4)^(2))/((x-2)(x^(4))) = 0. Since a fraction is equal to zero only if the numerator is equal to zero, we set the numerator equal to zero: (x+5)(x-4)^(2) = 0. Now, solve for x by setting each factor equal to zero: x+5 = 0 or x-4 = 0. This gives us x = -5 or x = 4. Therefore, the x-intercepts are (-5, 0) and (4, 0).
Step 4: Analyze the end behavior.
To determine the end behavior of the graph (i.e., how the graph behaves as x approaches positive or negative infinity), observe the highest-degree terms in the numerator and denominator. In this case, the highest degree in the numerator is 2 (from (x-4)^(2)) and in the denominator is 4 (from x^4). Since the degree of the numerator is less than the denominator, as x approaches positive or negative infinity, the function tends towards zero.
Step 5: Determine any vertical asymptotes.
Vertical asymptotes occur when the denominator of the function equals zero. In this case, x = 2 is a vertical asymptote because the function becomes undefined when x = 2.
Step 6: Plot additional points.
To get a more accurate graph, you can plot additional points by substituting various x-values into the function and calculating the corresponding y-values.
Step 7: Sketch the graph.
Using all the information gathered, including the domain, intercepts, end behavior, asymptotes, and additional points, you can sketch the graph of the function on a coordinate plane.
It is recommended to use graphing software or a graphing calculator to accurately plot the graph of this function, as it may be challenging to do so manually due to the complexity of the equation.