Use the guidelines given in Section 4.4 to make a complete gra...

Question

Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

ƒ(x) = (x⁴/2) - 3x² + 4x + 1

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Graph the following function on its domain. F of X is equal to x 4 divided by 16 minus X of 3 divided by 12 minus X2 + 3 X + 2. Awesome. So it appears for this particular problem we're asked to create a graph of this function of F of X based on its domain. So with that said, first off, we need to determine the domain of our particular function for this problem. So we need to note that since F of X is a polynomial, that will mean that its domain will be parenthesis negative infinity, comma. Infinity, and I'll put a capital P in quotation marks because this is because it's a polynomial. So this capital P denotes polynomial. So we need to note that since F of X is a polynomial, that will mean that its domain is within an open interval of negative infinity, comma infinity, because that's what the parentheses mean. Parentheses means it's an open interval. So that means our domain is open interval negative infinity positive infinity. So that's for our domain, which I'll say a capital D for domain. So now let's focus our attention on the symmetry, which I'll denote as S. So now we need to consider the symmetry. So as we should recall a note, in order to check the symmetry or to check for symmetry, 1 may compare F of negative X with F of X. So we need to consider F of negative X and F of X. However, in this particular case, a quick check will show that there is a presence of an odd power term because we have this negative X to the power 3 divided by 12, and we also have a linear term 3X, which will prevent the function from being even or odd. So therefore there is no Symmetry, so no S, so no symmetry. So I'll put S in quotation marks for symmetry. So there's no symmetry for this particular problem. And this is because there's a presence of an odd power term, which once again is the negative X of the power 3 divided by 12. We also have this linear term 3X, so that means that this will prevent the function from being even or odd, and due to this, there is no symmetry. So that was our, that's our 2nd step. So now our 3rd step now is we need to determine the first derivative to help us determine the critical points and the increasing and decreasing intervals. So we need to differentiate our function of F of X, so using the power. Rule of differentiation, we will find that F of X is equal to x to the power of 3 divided by 4 minus x to the power of 2 divided by 4 minus 2 x + 3. Fantastic. So now, Our next step is we need to set F of X equal to 0. So that means we need to take X 3 divided by 4, minus X 2 divided by 4, minus 2 X plus 3 is equal to 0. So now we need to multiply both sides by 4 to clear out our fractions. So when we do that, we will find that X to the power of 3, minus X2 minus 8 X plus 12 is equal to 0. Fantastic. So now we need to factor the polynomial X of the 3 minus X2 minus 8 X + 12, using our rational root theorem to help us find the possible rational roots. So as we should recall and note, the rational root theorem states that any rational root of a polynomial equation with an integer coefficient is a fraction. P divided by Q, where P is a factor of the constant term and Q is a factor of the leading coefficient. So for the polynomial x 3 minus X2 minus 8 X + 12, the constant term will be 12. And its factors will be plus or -1, + or minus 2, plus or minus 3, plus or minus 4, and we also need to note that it's gonna be plus or minus 6, and it will also be plus or minus 12. And the leading coefficient will be 1. And its factors will be plus or minus 1. Fantastic. So, with that said, Thus, as a result, therefore, the possible roots will be plus or minus 1, + or -2, plus or minus 3, plus or minus 4, plus or minus 6, and plus or minus 12. So we need to test these possible roots using synthetic division or substitution. Let us start with X equals 1, so X equals 1. Fantastic. So when X equals 1, that means we need to plug in a value of 1 in for X, when you take F of 1 is equal to 1 to the power of 3. So when you take 1. To the power of 3 minus 1 to the power of 2 minus 8 multiplied by 1 plus 12. And when we do that, we will find that it'll be equal to simply 4. And since F of 1, and we need to note, let's make a note that F1 cannot equal 0, so that means that X equals 1 is not a root. So let's try X equals 2, so X equals 2, so that will mean that F of 2. So instead of writing out the expression again, we'll just plug in a value 2 into our expression, and I'll just give you the answer. So when you plug in a value of 2 in for X, we will get an answer of 0. So that will mean that sense, we need to note that sense. F of 2. is equal to 0, since F 2 is equal to 0, that means that X equals 2 is a root. So now we can divide the polynomial by X minus 2. Fantastic. So using synthetic division, we will divide. X 3 minus X2 minus 8 X. Plus 12 by X minus 2. So when we do that, we will find the following we use synthetic division, so we'll say, So, 211-60, so we'll have 11, 8. And 12 And we'll say 22, and negative 12. So therefore, as a result, the quotient will be, so as we should determine, the quotient will be X2 + X minus 6. So X2 + X minus 6. So now we need to factor out X2 + X minus 6. So when we do that, we will find that it's gonna be when you factored out will be X + 3 multiplied by X minus 2. So thus, the factorization of X to the power of 3 minus X2 minus 8 X + 12 will be X minus 2 multiplied by X + 3 multiplied by X minus 2 will be all equal to X minus 2 to the power of 2 multiplied by X + 3. So now our next step is we need to take our expression for X 3 minus X2 minus 8 X + 12 equals 0, and we need to rearrange, let's write that down, so we need to take X 3 minus X2 minus 8 X + 12 equals 0, and we need to rearrange this equation to isolate. And solve for X all by itself. So when we do that, we will find that X is gonna be equal to simply -3, because as you can see, When we start to simplify, we'll get that X minus 2 to the power of 2 multiplied by X + 3 is equal to 0. So that will mean that we have X is equal to -3 and X is equal to 2. So there's two answers. Because as you can see, this X equals -3 comes from our one part, which is the X + 3 because X + 3 equals 0, that's how we get an X minus 0, so we're gonna get two separate answers, and then we find that X minus 2 to the power of 2 is going to give us X minus 2 equals 0, so then we'll get X equals 2. So let's Indicate those so we don't forget those. So those are our two answers for X. So therefore our critical points are gonna be X equals 2 and X equals -3. So, we'll make a note this is a CPR critical points. Fantastic. So now, we need to note that the first derivative can also be written as, let's make a note of this, that F of X is equal to 1/4 multiplied by X 3 minus X2 minus 8 X + 12 is equal to 1/4 multiplied by X minus 22 multiplied by X + 3. Fantastic. So, moving right along here, our next step now is we need to determine the increasing and decreasing intervals. So we need to note that when X is less than -3, that will mean that X minus 22 will be always positive. And that we need to note that X + 3 is going to be less than 0. So that will mean that F of X is less than 0, so that will mean that F is decreasing. So I'll put a capital D for decreasing. So now, and then we'll do an arrow pointing down, so we know for sure. So now, for -3 is less than X and X is less than 2, that will mean that X minus 2 to the power 2 is going to be greater than 0, and this will also mean that X + 3 is going to be greater than 0. So that will mean as a result that F of X is going to be greater than 0. So that will mean that F is going to be increasing. So I put a letter I for increasing and an arrow pointing up, so we know it's increasing. And we need to note that when X is greater than 2, Again, both factors will be positive, so that will mean that F of X is going to be greater than 0, so F will also be increasing. Fantastic. So therefore, without a doubt, at X equals -3, the derivative changes from negative to positive, so there will be a local minimum at X equals 2. The derivative will be zero, but without any sign change occurring. So therefore, that will mean that this is a critical point that does not correspond to a local extreme value. So now we need to calculate the function values at the critical points. So when X is equal to -3, that will mean that F of -3 is going to be equal to -3 to the power of 4. Divided by 16 minus -3 to the power of 3. Divided by 12 minus -3 2 plus 3 multiplied by -3 plus 2, which will be approximately equal to when we simplify to simplify our answer from a fraction to a decimal answer and rebound to two decimal places, we'll get that it's approximately equal to -8.69. Fantastic. So it's gonna be -8.69. Awesome. So now we need to focus on X equals 2. So when F of 2, so instead of writing out our function again, if we plug in a value of 2 in for X, we will get an answer that is approximately equal to 4.33. Fantastic. So now, we need to perform a 2nd derivative test. So we need to figure out the concavity and inflection points, so we need to differentiate F of X. So we have determined that F of X, so F of X, Is going to be equal to 1/4 multiplied by X minus 22 multiplied by X + 3. So we need to differentiate using the product rule so we could say that U of X is equal to X minus 22 and V of X is equal to X + 3. Fantastic. So then we can go ahead and say that U of X is equal to 2 multiplied by X minus 2, and V of X is equal to 1. So therefore, without a doubt, we can go ahead and write that. F. Of X is going to be equal to 14th multiplied by square bracket UX multiplied by VX. So UX multiplied by V of X, so UX multiplied by V of X plus U of X multiplied by VX. Which is gonna be all equal to 1/4 multiplied by square bracket of 2 multiplied by X minus 2. Multiplied by X + 3. Plus, X minus 2 to the power of 2, fantastic. So that means when we factor out X minus 2, we will find that F of X is going to be equal to 1/4 multiplied by X minus 2 multiplied by square bracket of 2 multiplied by X + 3 + X minus 2. So when we simplify even further, we will find that F. Of X is gonna be equal to 1/4 multiplied by X minus 2 multiplied by 2 X + 6 plus X minus 2, which is going to be equal to 14th multiplied by X minus 2 multiplied by 3 X + 4. Fantastic. So now, our next step is we're going to We need to find now the potential inflection points by setting F of X equal to 0. So if we take F of X is equal to 0, we will find that 14th multiplied by X minus 2 multiplied by 3 X. So 3 X plus 4 is equal to 0. So therefore our solutions are going to be when we solve for X, it's going to be X is equal to 2. And X is equal to -4 divided by 3, which is approximately equal to -1.33 when we round to two decimal places. And once again, that's when we rearrange our equation to isolate and solve for X, and we'll get two separate answers for X. So now we need to determine concavity. So in order to determine concavity, we need to test the intervals using F of X. So when X is less than -4 divided by 3, let us say that X is equal to -2. That will mean that X minus 2 is less than 0 and 3 X + 4 is equal to 3 multiplied by -2 + 4, which is equal to -6. +4, which is equal to -2, and 2 is less than 0. So the parentheses X minus 2 multiplied by parentheses 3 x + 4 is greater than 0, since a negative times a negative will give us a positive result. So that will mean that F of X is going to be greater than 0. So that means it's going to be concave up, which I'll call this Cu so it's gonna be concave up. Fantastic. And for the condition, negative 4 divided by 3 is less than X and X is less than 2. Let us say that X is equal to 0. So when X minus 2 is less than 0 and 3 X plus 4 is equal to 4, which is greater than 0, that will mean as a result, That Since we know that 3 X + 4 is going to be equal to 4 and 4 is going to be greater than 0, that will mean that the product is negative. So F of X is going to be less than 0, so that will mean for this condition it's going to be concave down, which I'll call CD. Fantastic. And for now, we need to say that when X is greater than 2, let us say that X is equal to 3, so that will mean that X minus 2 is greater than 0 and 3 X plus 4 is greater than 0. So that will mean that the product is positive, so F of X is going to be greater than 0, so it's going to be concave up. Fantastic. So, since there are changes in concavity at X equals -4 divided by 3 and X equals 2, that means that there will be inflection points present. And that will mean as a result that we need to find the Y coordinates of the inflection points, so that means we need to take F of -4 divided by 3 is equal to -4 divided by 3 to the power of 4 divided by 16. Minus -4 divided by 3 to the power of 3 divided by 12, fantastic. And then we need to take minus -4 divided by 3 to the power of 2, plus 3 multiplied by -4 divided by 3. Plus 2, so that will mean that we will get an answer of approximately 3.38 when we round to two decimal places. So that will mean that at X equals 2, we have already determined that F of 2 is approximately equal to 4.33. So, moving right along now to solving for our intercepts. So for Y intercepts, we will know that at X equals 0. And F of 0 equals 2, so that means that we will have a intercept on the Y axia at 0.2 in parentheses. As for our X intercepts, so let's say this is, this is Y. Intercepts Y I. So why I intercepts. So now we can say X intercepts. Awesome. I'll do a Y I for Y intercepts and an XI for X intercepts. So in order to find X intercepts, we need to find X such that F of X is equal to 0. So this is where F of X is equal to 0. So we need to solve the quadratic equation. X 4 divided by 16 minus X to the power of 3 divided by 12 minus X2 + 3 X + 2 is equal to 0, so we need to solve this quadratic function. Which is, which generally requires numerical methods or tedious factorization, but for curve sketching purposes, we do not need to solve this quadratic function. We will just need to utilize the other information that we have gathered to help us produce our graph. So with that said, let us focus on the asymptotes, which I'll denote as A. So focusing on our asymptotes, since F of X is a polynomial, there will be no Vertical or oblique or diagonal asymptotes. As for our end behavior, which I'll call EB end behavior, we need to note that as X approaches plus or minus infinity, The leading term X to the power of 4 divided by 16 dominates. So that will mean as a result that F of X will approach positive infinity. So that means we need to plot our local minimum at parentheses 3.8.69, and we also need to plot our inflection points, which is -4 divided by 3.3.38. And we need to also plot our other inflection point. Which we need to note once again is 2.4.33, and we also need to note that our Y intercept is going to be at 0.2, and we need to connect these points with a smooth curve and using our reference to increasing and decreasing intervals, concavity and end behavior. And when we consider all this information that we've assembled, We will be able to go and finally officially graph our answer, so we can finally create our final graph. So with that said, Considering everything that we just solved together, we should be able to produce the following graph. So with that said, Our graph should look as follows. So, as we can see, we have our different points that we plotted. So we have 4 different points. So we have our -3.8.69. Then we have our -4 divided by 3.3.38. Then we have our 0.2, which is our Y intercept, the only intercept that we have where it crosses our Y axis. And then we have 2.4.33, and as you can see when we connect it with a smooth curve, we can see that it's concave up. And we can see where it's concave up, we could see where it's decreasing, we could see where it's increasing, we can see where where it's concave down and concave up, and as you could see, it goes down, that's decreasing, and then it forms like a parabolic shape, and then it'll start to increase, increase, increase, and then it'll decrease slightly, and then it'll start to increase again. Fantastic. And that's it. We've officially solved for this problem. Thank you so much for watching. So this problem is a little bit complicated, but once you have a strong understanding of graphing techniques, it's going to be very easy to solve. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = (x⁴/2) - 3x² + 4x + 1

Key Concepts

Polynomial Functions

Graphing Techniques

Using Graphing Utilities

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