49–54. {Use of Tech} Graphing with technology Make a complete ...

Question

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.

ƒ(x) = 1/3 x³ - 2x² - 5x + 2

Accepted Answer

A little there. Today we're going to 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 function F of X is equal to x the power of 3 divided by 4 minus 3 x 2 minus 6 X plus 12. Awesome. So it appears for this particular problem we're ultimately trying to. Figure out how to graph this function of F of X. So now that we know that we're trying to grab like graph the specific function of F of X, first off, let us follow a step by step method to help us analyze and plot our graph for the function. F of X is equal to x to the power of 3. Divided by 4. Minus 3 X to the power of 2, minus 6 X plus 12. Fantastic. So now our first major step towards solving this particular problem is we need to identify the domain. So as we should recall a note, since F of X is a polynomial function, it is defined. For all real numbers, meaning the domain will be negative infinity positive infinity within parentheses where parentheses indicate an open interval, whereas square brackets will denote a closed interval. So once again, because we're dealing with a polynomial function, that means that it's defined by all real numbers, and that's why we could say parenthesis negative infinity, comma infinity for our domain. Our second step now is we're asked to check for the symmetry. So, as we should recall, in order to check for the symmetry, we need to test whether F of negative X is equal to F of X, which is an even function. So let's make a note of that with a capital E for even or if it is F, so F of negative X. is equal to negative F of X, and this F of negative X is equal to negative F of X will be an odd function, so E for even, O for odd for short. So with that said, let us compute F of negative X. So when we do that, we will find that F of negative X is equal to the power of 3 divided by 4 minus -3 multiplied by X to the power of 2. Minus -6 multiplied by negative X plus 12, which will mean that F of negative X will be simply equal to when we simplify negative X to the power of 3 divided by 4 minus 3. X to the power of 2 plus. 6 X plus 12. Fantastic. So, since F of negative X cannot equal positive, which I'll get rid of the positive symbol, so it cannot equal positive F of X, and that F of negative X is cannot equal. So we're also saying that F of negative X cannot equal positive F of X and F of negative X cannot equal negative F of X. That means that the function is neither odd or even. So it's neither. So I'll put a giant letter N for neither, so it's neither odd nor even. Awesome. So, our 3rd step now. we need to find the 1st and 2nd derivatives. So the first derivative, which will be F of X, so F of X will be equal to 3/4 x 2, so 3/4 x 2 minus 6 X minus 6, and solving for our second derivative, which will be F of X and F of X will be equal to 64 64 6 divided by 4 X minus 6. Which will be equal to and we do a little bit of reducing and writing in its most simple form. 3 halves X minus 6. So now our 4th step is we need to find the critical point, or critical point or points. So as we should recall, the critical points occur when F of X is equal to 0, so we need to take 3/4 x 2 minus 6 X 6 and set it equal to 0. So now what we're gonna be doing is we're going to be rearranging this expression to isolate and solve for X. So in an effort to save some time, since it's going to be basic algebra, we're rearranging our expression to isolate and solve for X, which, as you can see, we're dealing with a quadratic formula, so you're gonna have to recall and use the quadratic function. So, so essentially you could simplify this out first by multiplying by 4 to clear all the fra fractions. So when you do that, you will get 3 X to the power 2 minus 24 X minus 24, so 24 X minus 24 is equal to 0. And then when we recall and use the quadratic function, we will find that X is equal to negative, -24 plus or minus the square root of -24 to the 2 minus 4 multiplied by 3 multiplied by -24. All divided by 2 multiplied by 3. So when we solve for X, we will find that X is approximately equal to 8.9 and X will also be approximately equal to -0.9. Fantastic. So that will mean that our critical points are 8.9, or rather X is approximately equal to 8.9, and X is approximately equal to -0.9, fantastic. So, moving right along here, our 5th step now is we need to determine the increasing and decreasing intervals. So we need to use the sign of F of X. So we need to choose our test points within our interval. So let us focus on the interval negative infinity -0.9 in parentheses and 0.98.9 in parentheses and 8. Positive, so to be precise, 8.9, infinity. Fantastic. So, and don't forget once again that these are in parentheses indicating an open interval. So now we need to test our intervals. So with that said, We need to test X is less than -0.9 1st. So let us test when X is equal to -1. So in F of -1, so when X is equal to -1, it will be equal to 0.75, which is greater than 0, which will mean that it is increasing. For the condition where -0.9 is less than X and X is less than 8.9, let us test X is equal to 0. So F of 0, so we're plugging in a value of 0 into our F of X expression. So when we plug in a value of 0, you will find that it's equal to -6, which is less than 0, which means that it will be decreasing. And with that said, our third interval now is for when X is greater than 8.9. So let us test when X is equal to 9 in this case. So F of 9 is going to be equal to 0.75. Once again, we're plugging in a value of 9 in for X, which this value is greater than 0, which means that it's going to be increasing, fantastic. So now we need to evaluate the derivative. So when we evaluating the derivative, we can determine that F of X is increasing. On the intervals negative infinity 0.9 and 8.9, infinity. So that's increasing on those two intervals, and F of X is decreasing on the interval -0.9.8.9 in parentheses. Awesome. So moving right along here. So our 6th step is we need to find the concavity and the inflection points. So in order to do that, as we should recall, we need to find F of X is equal to 0, so we need to take 3 halves X minus 6, so 3 halves X minus 6 is equal to 0. So, we need to rearrange this equation to isolate and solve for X by itself, and when we do that, using a little bit of algebra, we will find that X is equal to 4. So that means that X is equal to 4 is an inflection point, which I'll call IP for short. So now we need to test our concavity. So, for the condition X is less than 4, let us test X is equal to 0. So F of 0 is equal to 3 multiplied by 0, divided by 2 minus 6, which is equal to -6, which is less than 0, which will mean that it's concave down. For when X is greater than 4, let us test where X is going to be equal to 5 now for this particular case. So instead of rewriting our equation out again, I'll just plug in the value of F of 5 in tor expression, since we're plugging in a value of 5 in for X. And when we do that, we will find that F of 5 will be equal to 1.5, which once again is greater than. So, once again, we said that -6 is less than 0, so it's concave down. However, 1.5 is greater than 0. So that means it's going to be concave up in this particular case. So, by evaluating the second derivative, we can conclude that F of X is concave down on, so it's going to be concave down on the interval, negative infinity 4 in parenthesis. And drum roll, and it's gonna be concave up on parentheses, for infinity in parentheses. So, therefore, there is an inflection point at Drum roll parentheses 4. F of 4 or at 4.44. Fantastic. So with that said, moving right along here, so our next step now, this is step 7. So for step 7, we need to find our intercepts. So let's start with the Y intercept first. So we need to set X equal to 0. So when we do that, we will find that F of 0, which we're plugging in a value of 0 in for X for our original function. So when we do that, we'll find that F is 0 is equal to 0 to the power of 3, divided by 4 minus 3 multiplied by 0 to the power 2 minus 6 multiplied by 0 + 12, which will mean that F of 0 is going to be equal to simply 12. So that will mean that our intercept for the Y intercept will be at 0.12 in parenthesis. Fantastic. So now to solve. For our X intercept, so solving for our X intercept we need to note that to do that, as we should recall, we need to take F of X is equal to 0, so we set Y equal to 0, so we will take X to the power of 3, divided by 4 minus 3, X2 minus 6, X plus 12 is equal to 0. So when we solve numerically, we will find that our intercepts for the X intercepts are going to be at parentheses negative 2.8.0 and 1.3.0, and finally, lastly, 13.50. Fantastic. So, now, moving right along here for our eight step. We need to figure out the end behavior, so we need to note that as X approaches infinity, the highest degree of the term X 3 divided by 4, dominates. So that will mean that F of X will approach infinity. So once again, F of X will approach infinity. And we need to note that since F of X approaches infinity, that will mean that as X approaches negative infinity, that will mean that, so as X approaches negative infinity, since X to the power of 3 is negative. For large negative x values, we will find that F of X will also approach negative infinity. So that means in conclusion, the function will be increasing on parentheses negative infinity, 0.9 and 8.9 infinity and decreasing on parentheses 0.9.8.9. It will have local extrema at X is approximately equal to -0.9, and X is approximately equal to 8.9. It also has an inflection point at X equals 4, and there's X intercepts at 2.8, 1.3.0 and 13.50. There's a Y intercept at 0.12. This particular function does not have symmetry, and the end behavior is as F of X approaches infinity as X approaches infinity and F of X approaches negative infinity as X approaches negative infinity. So knowing all this information that we've collected together, we can officially plot all of our information onto our graphs. We're going to plot all of our coordinate points that we found together and then connect them with a smooth curve. So I've gone ahead and plugged our function for FFX into some graphing software to produce the following graph. As you can see, we have our curve represented in blue, and all of our coordinate points are represented by red dots. We have our local minimum at X equals 8.9. Then we have our inflection point at 4.44. And then we have our different coordinate points at -2.8.0, 0.12, 1.3.0, and 13.5.0. And we also have our local maximum at X equals -0.9. Awesome. And as you can see, our curve starts in the negative 3 quadrant, and it goes up, forms an inverted parabola shape, dips down, and then it forms an upright parabola shape, and then curves upwards into our first positive quadrant. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.

ƒ(x) = 1/3 x³ - 2x² - 5x + 2

Key Concepts

Graphing Functions

Intercepts

Local Extreme Values and Inflection Points

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