Determine all significant features (approximately if necessary) and sketch a graph. f(x)=x √(x^2

Determine all significant features (approximately if...

Key Concepts

Critical Points and Derivative Analysis

Analyzing the derivative of a function helps identify critical points where the function's slope is zero or undefined. These points often correspond to local extrema or inflection points. Understanding these features is key to determining the overall shape and turning behavior of the graph.

Domain Analysis

Determining the domain of a function is crucial when dealing with expressions like square roots. For a function with a square root, the expression inside the root must be non?negative. This concept ensures that the function is defined over real numbers and sets the stage for further analysis of its behavior.

Symmetry Analysis

Checking a function's symmetry involves testing whether the function is even, odd, or neither. An odd function, where f(?x) = ?f(x), implies symmetry with respect to the origin. This insight can simplify the analysis and sketching of the graph by reducing the workload needed to understand the function’s behavior on the entire domain.

Intercepts Calculation

Identifying intercepts involves finding where the function crosses the coordinate axes. The x-intercepts are found by solving f(x) = 0, while the y-intercept is obtained by evaluating the function at x = 0 (if it lies in the domain). Knowing the intercepts helps in anchoring the graph accurately on the coordinate plane.

End Behavior and Asymptotic Analysis

Examining the function's behavior as x approaches positive or negative infinity reveals its end behavior. This analysis often shows if the function approaches a line or tends to infinity, which is essential for accurately sketching the graph and understanding the limits at far ends of the domain.

Graph Sketching

Graph sketching is the synthesis of domain analysis, symmetry, intercepts, critical points, and end behavior. This comprehensive approach allows one to construct an accurate representation of the function. The combined insights result in a qualitative picture of the graph, highlighting key features and overall trends.

Transcript

00:01 For this problem, we've been asked to find all the significant features of the function f of x equals x times the square root of x squared minus 4, and then try to sketch the graph.

00:12 So first, let's examine the function itself.

00:15 I have a square root here.

00:16 So that's going to put a limit on the domain for this function.

00:19 We need to have whatever's under the radical be greater than or equal to 0, which means that x squared has to be greater than or equal to 4, or x is either greater than or equal to 2 or x is less than are equal to negative 2.

00:36 Here's my domain.

00:37 So i've got two sections of this domain, and there is a gap between negative 2 and 2, where this function is undefined.

00:45 And i'd also like to know where the zeros of this function are.

00:48 Well, x equaling 0, well, it looks like a 0, it is not actually part of our domain, so that is not going to be something we'll look at for our graph.

00:59 And the other ones is when what's under the radical equals 0.

01:02 And that happens when x is 2 and negative 2.

01:08 So at the endpoints of each of these sections of the domain are my zeros.

01:14 Okay, now let's work on finding the critical points, which is going to be my first derivative.

01:20 I'm going to take this function, and i'm going to just rewrite it with an exponent instead of a radical.

01:27 That makes it a little bit easier to use our derivative rules.

01:31 This is a product, so i'm going to use the product rule to find the first derivative.

01:34 So the first times the derivative of the second.

01:39 That's going to be one half times x squared minus four to the negative one half times the derivative of what's in the parentheses, 2x.

01:48 So first times the derivative of the second plus the second times the derivative of the first.

01:58 Let's clean this up a little bit.

02:00 If i combine and cancel as much as i can, my first term, well, the two and the one half cancel, my first piece is, is x squared times x squared minus 4 to the negative 1 half.

02:12 And then i end up with x squared minus 4 to the 1 half in my second term.

02:19 Factor out everything i can.

02:22 So i'm gonna factor out an x squared minus 4 to the negative 1 half.

02:27 When i do that, i'll end up with x squared plus an x squared minus 4 to the first power.

02:36 And i'm just gonna combine everything i can within that second set of parentheses.

02:40 And i get 2x squared minus 4.

02:47 Now, critical points.

02:49 When is this equal to 0? well, this first term is actually undefined at my endpoints, at my 0s, when x equals 2 or negative 2.

03:00 So there are nothing new.

03:01 I already have those down as pieces of information.

03:05 I also have another term.

03:07 When is that equal to 0? well, if i take 2x squared minus 4 and set it equal to 0, that gives me x squared minus 4 and set it equal to 0, squared equals 2.

03:18 But when i solve this, the square root of 2, both of these are within that section that is not included in my domain.

03:25 Square root of 2 is about 1 .732.

03:28 These do not fit in my domain.

03:30 So i actually have no critical points.

03:41 Okay.

03:42 Now what about second derivative? what about inflection points? well, i'm going to come up so i have a little bit more room.

03:49 And let's find the second derivative.

03:52 Again, i have a product.

03:54 So it's going to be the first times the derivative of the second, which is going to be 4x, plus the second times the derivative of the first.

04:10 So that's going to be negative 1 half times x squared minus 4 to the negative 3 halves, times the derivative of what's in the parentheses, 2x.

04:23 Okay, we're going to clean up as much as we can.

04:25 First term is pretty straightforward.

04:29 Nothing really combines at this point...

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