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

For each polynomial function, find (a) the end behavior; (b) the $y$ -intercept; (c) the $x$ -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=-3(x-2)^{2}(x+1)^{2}$$

Key Concepts

Factoring Polynomial Functions

Factoring a polynomial into its component linear or irreducible quadratic factors is a key technique used to reveal the x-intercepts and their multiplicities. This process simplifies computations and helps in identifying the roots and behavior of the function near those roots.

Graph Sketching

Graph sketching compiles all the analyzed information—end behavior, intercepts, symmetry, and positivity/negativity intervals—to produce an accurate representation of the function’s behavior. A coherent graph demonstrates how these characteristics interact to shape the overall curve of the polynomial function.

Intervals of Positivity and Negativity

To find the intervals where a polynomial function is positive or negative, one must evaluate the sign of the function in the intervals between the x-intercepts. This involves testing sample points from each interval. Understanding these intervals provides insight into where the function is above or below the x-axis, which is critical for accurately sketching the overall graph.

End Behavior

The end behavior of a polynomial function is determined by its leading term, meaning the term with the highest degree. This concept entails looking at both the degree and the coefficient of that term to decide how the function behaves as x approaches positive or negative infinity. In general, if the degree is even, the ends will either both point upward or downward depending on the sign of the leading coefficient; if the degree is odd, the two ends point in opposite directions.

x-Intercepts and Multiplicities

Finding the x-intercepts involves solving the equation f(x) = 0, which typically requires factoring the polynomial and setting each factor equal to zero. The multiplicity of each zero indicates the behavior of the graph at the corresponding x-intercept. A zero with an odd multiplicity means the graph crosses the axis, whereas a zero with an even multiplicity means the graph touches the axis and turns around.

y-Intercept

The y-intercept of a polynomial function is found by evaluating the function at x = 0. This value shows where the graph intersects the y-axis and is essential for graphing as it provides one fixed point of the function.

Graph Symmetry

Determining symmetry in a polynomial function is a method of finding if the function is even, odd, or neither. An even function satisfies f(-x) = f(x) and is symmetric about the y-axis, while an odd function satisfies f(-x) = -f(x) and is symmetric about the origin. Recognizing symmetry can simplify the graphing process by reducing the amount of computation needed.

Transcript

00:01 We are asked to sketch a function again, and we know how to first find the boundary.

00:10 So we know that when x approaches infinity, when x approaches infinity, our f of x, our f of x is equal to, or our f of x approaches.

00:28 So let's look at it.

00:29 So when we have, since we have these two terms that are squared, this x minus two squared, and x plus 1 squared, those terms will always be positive.

00:40 And so the negative 3, the leading coefficient of negative 3, will always make this negative.

00:47 So this will also be negative infinity.

00:49 And when x approaches negative infinity, negative infinity, f of x will also approach negative infinity, because the two terms that are squared will always be positive.

01:04 So we will always have a positive number times a negative number, which will make it negative.

01:10 And so since we have the boundaries, we can now look at the y intercept.

01:18 And so to find the y intercept, we set x to be 0.

01:22 So 0 minus 2 squared times 0 plus 1 squared, which is equal to negative 3 times 4.

01:34 Times 1.

01:36 So negative 3 times 4 is negative 12.

01:41 So now we can look at our x intercepts.

01:47 And those are x is equal to 2 and x is equal to negative 1.

01:52 And both of these, both of these zeros have a multiplicity of 2.

01:58 And so what this means is that they will only touch, they will only touch the x -axis.

02:10 So now we can look at, we can look at symmetry.

02:16 So we know that we have f of x, f of x is equal to negative f if f of x is equal to f negative x, then we have an even function.

02:33 And if f of x is equal to negative f of negative x, then we have an odd function.

02:44 And so let's let's see if we have an even - our odd function.

02:47 So we have negative 3x minus 2 squared x plus 1.

02:57 And so this is equal to negative 3 times negative x minus 2 squared times negative x plus 1.

03:11 And so right away we can see that these inside terms are different.

03:15 So we have negative x minus 2 compared to x minus 2 and negative x plus 1 compared to x plus 1...

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