For each polynomial function, find (a) the end behavior; (b) the y -intercept; (c) the x -interc

step 1 to sketch the given a function based off of what we know. So first thing we want to do is we wan

For each polynomial function, find (a) the end behavior; (b)...

Key Concepts

Graph Sketching of Polynomial Functions

Graph sketching integrates all of the above concepts – end behavior, intercepts, symmetry, zeros with their multiplicities, and sign intervals – to provide a complete picture of the polynomial's graph. Understanding each of these components allows one to produce an accurate visual representation of the function, capturing its essential characteristics without needing to plot every point.

Symmetry in Polynomial Functions

Symmetry is used to simplify the analysis of the graph of a function. A function may be even, odd, or neither, which can be determined by testing the function for symmetry about the y-axis or the origin. Recognizing symmetry can reduce the amount of work needed by allowing parts of the graph to be reflected appropriately.

Interval Testing for Sign Analysis

Sign analysis involves determining the intervals on which the polynomial function is positive or negative. This is done by finding the zeros, segmenting the number line, and then testing points from each interval. This process helps in understanding where the graph lies above or below the x-axis and is crucial for accurate graph sketching.

Y-intercepts of Polynomial Functions

The y-intercept is the point where the graph of the function crosses the y-axis, which is found by evaluating the function at x = 0. It provides an initial anchor point for graphing the function and plays an important role in understanding the overall behavior of the graph.

Polynomial End Behavior

The end behavior of a polynomial function is determined by its leading term, specifically the degree and the leading coefficient. This concept explains how the function behaves as x approaches positive or negative infinity. For example, if the degree is even and the leading coefficient is negative, the function will tend to negative infinity on both ends, whereas if the degree is odd, each end will behave differently based on the sign of the leading coefficient.

Zeros and Their Multiplicities

Finding the zeros (or x-intercepts) of a polynomial involves solving the equation set equal to zero. Factoring the polynomial is often used to determine these intercepts. The multiplicity of a zero indicates how many times a particular solution is repeated and influences the graph's shape at that intercept (e.g., whether the graph crosses the axis or merely touches it and turns around).

Transcript

00:01 To sketch the given a function based off of what we know.

00:05 So first thing we want to do is we want to factor the function.

00:10 So we see that we have an x squared for all of our terms.

00:15 So let's say, so we have, or let's, let's factor out a negative x squared.

00:21 So we have negative x squared.

00:23 So we're left with x squared.

00:27 It's nice that we have a positive one now.

00:28 X squared let's factor out a negative 2 x squared because we have we have multiples of 2 so we have negative 2 x squared times now we have an x squared so this is nice because our a values 1 now negative so plus my sorry minus 2 x plus sorry minus x plus sorry minus again and we have minus minus x minus 1 because x squared times negative 2x let's check out what x squared times negative 2 x is negative 2x to the 4th which is correct we have negative 2x times negative 2x squared which is 4x cubed and then negative 1 times negative 2x squared is 2x squared so that that is correct and so now we can see if we have so let's say we have b is b is negative 2 and c is negative 1 and so we want two numbers that that add up to negative 2 but multiply to negative 1 and so since there are no two numbers that that add up to negative 2 but multiply to negative 1 so we have to use a quadratic formula for this so let's do negative b which is 2 negative b plus for minus the square root of b squared which is four minus four times one four times another one so four plus four over two and so what this gives us is now that we have so this this root eight is the same thing as saying is the same thing as saying square of four times squared of two and so the square of four is two this this is equal to the two root two and so if we divide by 2, we are left with 1 plus or minus root 2.

02:49 And so remember, since this has a multiplicity of 1, each of these roots, 1 plus root 2 and 1 minus root 2, they will have multiplicities of 1.

03:03 And so now that we have that, so this is our final function that is completely full.

03:14 Factored.

03:16 So now we can see if we have, now we can see what we have.

03:24 So we can find our boundary by saying, okay, if x approaches infinity, if x approaches infinity, well, this infinity term, so we have to look at our leading coefficient.

03:36 And so in our original problem, our leading coefficient was, was negative 2.

03:44 And that was on the variable with the highest exponent.

03:49 So in this case we had x to the fourth and our leading coefficient was negative.

03:53 So that means when when x is infinity, this will become x x x.

04:10 And so it's the same thing for x is negative infinity.

04:21 Because we have the x to the fourth, with this, the negative affinity or the infinity will always be positive.

04:29 But since we have that negative 2 as a leading coefficient, whatever we get will always be negative.

04:41 So as x approaches infinity, we get f of x approaches negative affinity.

04:47 And as x approaches negative infinity, we get f of x approaches negative infinity.

04:51 And all of that has to do with the leading coefficient of negative 2 on the variable with the highest x 1.

04:57 So now we can find our x intercept.

05:04 And so since we don't have a constant, we're going to have all of our terms.

05:08 So let's do negative 2 times 0 to the 4 plus 4 times 0 cubed plus 2 times 0 squared.

05:21 So this is 0.

05:26 Now we can look at our symmetries.

05:30 And remember if we have f of x, is equal to f of negative x this function will be even if we have f of x is equal to the negative f of negative x this function will be odd and so let's see if we have an even or odd function so we have negative 2x to the 4th plus 4x plus 2x plus 2x squared and so we want to make all of these x values now so we have negative 2 negative x to the 4 plus 4 negative x cubed plus 2 tons negative x squared and so we notice that okay these and these are the same because it's x to the 4th and x squared we are raising them to even powers so they'll just be x to the 4th and x squared but this this is what makes the difference...

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