Question 405036: My algebra assignment asks me to calculate, graph and answer in essay form, the following equation:
Graph: f(x) = 3x^3 + x^2 - 12x - 4
I am having difficulty understanding the steps in working these problems, let alone graphing them. I would appreciate any assistance you can provide.
Thanks, Duane
Found 2 solutions by lwsshak3, ewatrrr:
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Graph: f(x) = 3x^3 + x^2 - 12x - 4
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To graph this function, you need to know the y-intercept and x-intercepts, also known as roots or zeros.It is easy to find the y-intercept. Just set x=0, and you can see that the y-intercept=-4.
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Since it is a 3rd degree equation, we do not have a formula like the quadratic equation that finds roots for 2nd degree equations. In a 3rd or higher degree equation,if a rational root exists and we can find it, we can find the other remaining roots. There is a method to identify all the possible rational roots, and by a process of elimination a root could be found, but this is a long and relatively combersome process.
If you are permitted, the best way to find out if a rational root exists is to use a graphing calculator. For the given equation, you will find that 2 is one of the roots. To find the other two roots, just divide f(x) by (x-2) either by long division or synthetic division. You will find that the quotient will come out to 3x^2+7x-2. This will then factor to (3x+1)(x+2). In factored form, f(x)=(x+2)(3x+1)(x-2). Zeros are therefore, -2,-1/3, and 2.
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Now, for the graph:
On the y-axis mark the point, -4. On the x-axis mark the x-intercepts or zeros found. (At these points, f(x)=0. This is why the are called zeros. ) The graph (not to scale) must connect these marks. In order to do this right, we must determine whether f(x) is positive or negative between the zeros. Note that for numbers >2,that is, the interval (2,+infinity), f(x) is positive, so the curve will appear to be above the x-axis. Going from right to left, the interval between 2 and -1/3 will be found to be negative and the curve appears below the x-axis. The interval between -1/3 and -2 will be found to be positive and the curve appears above the x-axis. From here on, <-2, the curve stays below the x-axis. How do we determine whether the interval is positive or negative? Look at f(x) in factored form above and imagine a very large number >2. You can see that f(x) is positive in this interval. As you move from right to left thru the other zeros, signs of the subsequent intervals switch if the zero is an odd multiple (1,3,5,etc), and will not change if the zero is an even multiple (2,4,6,etc). I this case the zeros are of multiple 1, so the intervals switch signs.
The graph you draw should be similar to the graph below except usually you do not need to draw it to scale. Just mark the y-intercept and show whether the curve is above or below the x-axis.
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Answer by ewatrrr(24785)  (Show Source):
You can put this solution on YOUR website!
Hi
f(x) = 3x^3 + x^2 - 12x - 4
f(x) = (x-2)(3x^2 +7x +2)
f(x) = (x-2)(3x+1)(x+2)
Roots
(x-2)(3x+1)(x+2)= 0
(x-2)= 0 x = 2
(3x+1)=0 x = -1/3
(x+2)= 0 x = -2
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