Problem 9 Sketch a graph of the function s... [FREE SOLUTION]

Chapter 0: Problem 9

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=x^{4}-1$$

Short Answer

Expert verified

The function $f(x)=x^{4}-1$ has x-intercepts at $x=-1$ and $x=1$, and y-intercept at $(0, -1)$. There are no asymptotes and it has a local minimum at $x=0$. The graph of the function is an upward opening even function with a pinch point at $(0,-1)$.

Step by step solution

Find the intercepts

To find the x-intercepts, set $f(x) = 0$ and solve for x: $x^{4}-1 = 0 \Rightarrow x^{4} = 1$. This gives $x = \sqrt[4]{1}$ and $x = -\sqrt[4]{1}$, so the x-intercepts are $x=-1$ and $x=1$. For y-intercept, set $x=0$, then $f(0) = (0)^{4}-1 = -1$, so the y-intercept is $(0, -1)$.

Find the extremes

To find the extreme points, take the derivative of the function $f'(x) = 4x^{3}$, and set it equal to 0 to find the critical points: $4x^{3} = 0 \Rightarrow x = 0$. Since this is a even function, the only critical point at $x=0$ leads to a local minimum, not a local maximum.

Plot the function

Now, we have all information needed to sketch the graph of the function which includes the y-intercept at $(0, -1)$, x-intercepts at $(-1, 0)$ and $(1, 0)$, and there is a local minimum at $(0, -1)$. The function resembles an upward-opening even function with pinch point at (0,-1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number powers and coefficients. These functions can be as simple as a constant or as complex as high-degree polynomials. Each term's exponent or the degree of the polynomial provides insights into its properties. In this case, we are dealing with a function, $f(x) = x^4 - 1$, which is a fourth-degree polynomial.

The general form of a polynomial is $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0$
Polynomial graphs exhibit curves and can have multiple turning points.
The degree of the polynomial indicates the maximum number of x-intercepts it can have.

For $f(x) = x^4 - 1$, this means the graph is symmetric about the y-axis because all exponents are even. This symmetry results in a characteristic shape that resembles a 'W' or 'U', depending on the leading coefficient sign.

Extreme Points

Extreme points are locations on a graph where a function reaches a local maximum or minimum. In the case of $f(x) = x^4 - 1$, we can determine the extreme points by using calculus.Taking the derivative of the function, $f'(x) = 4x^3$, allows us to analyze where the slope of the graph is zero, indicating potential extreme points. When $f'(x) = 0$, we solve to find:

\[4x^3 = 0 \Rightarrow x = 0\]

This critical point at $x = 0$ is where the function's slope changes. Observing the graph, this leads to a local minimum at the point $(0, -1)$. Since the derivative does not change signs around $x = 0$, there are no local maxima present in this function.

Intercepts and Asymptotes

Intercepts are points where the graph crosses the x-axis or y-axis. Finding intercepts is crucial for understanding a function's behavior and overall shape.

X-Intercepts: Set $f(x) = 0$. For our function, $x^4 - 1 = 0$ yields $x = \pm 1$. Therefore, the graph crosses the x-axis at points $(-1, 0)$ and $(1, 0)$.
Y-Intercept: Set $x = 0$. For the function, $f(0) = -1$, so the graph crosses the y-axis at $(0, -1)$.

Asymptotes are lines that a graph approaches but never touches. Polynomial functions do not have vertical or horizontal asymptotes because they tend to infinity as $x$ becomes very large or very small. Nevertheless, polynomial growth rate and sign of the highest degree term guide the end behavior.

Derivative Analysis

Derivative analysis involves examining the derivatives of a function to comprehend its behavior and predict the graph's shape. Through derivatives, we can determine slopes, identify critical points, and analyze concavity.For $f(x) = x^4 - 1$:

First Derivative, $f'(x) = 4x^3$: Helps find critical points. At $x = 0$, the derivative is zero, indicating potential extremum.
Second Derivative, $f''(x) = 12x^2$: Indicates function's concavity. Since $f''(x) > 0$ for all $x$, the graph is concave up everywhere, confirming the presence of a minimum and no points of inflection.

Derivative analysis allows us to understand the function's behavior in detail, supporting us in sketching accurate graphs and predicting how polynomial functions behave across their domains.

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91影视

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=x^{4}-1$$

Short Answer

Step by step solution

Find the intercepts

Find the extremes

Plot the function

Key Concepts

Polynomial Functions

Extreme Points

Intercepts and Asymptotes

Derivative Analysis

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