MTH120 College Algebra Chapter 5.2

 5.2 Power Functions and Polynomial Functions:

Power functions and polynomial functions are fundamental types of mathematical functions that are used to model a wide range of phenomena in various fields, including mathematics, science, engineering, and economics. Let's explore these functions:

Power Functions:

A power function is a type of mathematical function defined by the equation:

f(x) = ax^n

Where:

• f(x) is the dependent variable or the output of the function.
• x is the independent variable or the input of the function.
• a is a constant coefficient.
• n is a constant exponent, which can be any real number.

Key points about power functions:

• The exponent n can be positive, negative, or zero, which results in various types of power functions.
• When n is a positive integer, the function describes a power-law relationship, where the output varies with the nth power of the input.
• When n is a fraction (e.g., 1/2, 1/3), the function represents a root function, such as square root or cube root.
• When n is negative, the function represents an inverse relationship, where the output decreases as the input increases.
• Power functions are commonly used to model phenomena where one quantity is proportional to a power of another quantity, such as distance vs. time in uniformly accelerated motion.

Polynomial Functions:

A polynomial function is a type of mathematical function defined by the equation:

f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0

Where:

• f(x) is the dependent variable or the output of the function.
• x is the independent variable or the input of the function.
• a_n, a_{n-1}, ..., a_2, a_1, a_0 are constants called coefficients.
• n is a non-negative integer called the degree of the polynomial. It represents the highest power of x in the function.

Key points about polynomial functions:

• The degree of a polynomial determines its behavior and the number of solutions to certain equations.
• The term with the highest power of x (the leading term) often dominates the behavior of the polynomial for large values of x.
• Polynomial functions can have real and/or complex roots (solutions) depending on their degree.
• Common types of polynomial functions include linear functions (degree 1), quadratic functions (degree 2), cubic functions (degree 3), and higher-degree polynomials.
• Polynomial functions are used in various fields for curve fitting, interpolation, and approximation of data.

In summary, power functions and polynomial functions are essential mathematical tools used to describe and model relationships between variables in many areas of science and engineering. They have diverse applications and play a crucial role in understanding and predicting real-world phenomena.

Power functions are mathematical functions defined by equations of the form:

f(x) = ax^n

Where:

• f(x) is the dependent variable or the output of the function.
• x is the independent variable or the input of the function.
• a is a constant coefficient.
• n is a constant exponent, which can be any real number.

To identify a power function, you need to look for the following characteristics:

1. Single Term: Power functions consist of a single term with the independent variable x raised to a constant exponent n. There are no addition or subtraction operations between terms.

2. Constant Coefficient: There is a constant coefficient a multiplying the term with x^n. This coefficient can be any real number, including zero.

3. Exponent n: The exponent n can be any real number, which distinguishes power functions from polynomial functions where n must be a non-negative integer.

4. No Other Terms: There are no other terms with different exponents of x in the equation. A power function has only one term.

Here are some examples of power functions:

• f(x) = 5x^3 (a power function with a positive integer exponent).
• f(x) = 2x^(-1.5) (a power function with a negative non-integer exponent).
• f(x) = 4x^0.5 (a power function with a fractional exponent).

These examples meet all the criteria mentioned above and are considered power functions. Power functions can describe various mathematical relationships and are widely used in science and engineering to model different phenomena.

The end behavior of a power function refers to how the function behaves as the input variable x approaches positive infinity (+∞) and negative infinity (-∞). Understanding the end behavior is essential for characterizing the long-term behavior of a function. It helps determine whether the function grows without bound, approaches a finite limit, or oscillates.

The end behavior of a power function f(x) = ax^n depends on the values of the coefficients a and n. Here are the rules for identifying the end behavior of power functions based on the values of a and n:

1. When n is an Even Integer:

   • If a > 0, as x → +∞, the function approaches positive infinity, and as x → -∞, the function approaches positive infinity.
   • If a < 0, as x → +∞, the function approaches negative infinity, and as x → -∞, the function approaches negative infinity.

   Examples:

   • For f(x) = 2x^2, as x becomes very large (positive or negative), f(x) becomes very large and positive.
   • For f(x) = -3x^4, as x becomes very large (positive or negative), f(x) becomes very large and negative.

2. When n is an Odd Integer:

   • If a > 0, as x → +∞, the function approaches positive infinity, and as x → -∞, the function approaches negative infinity.
   • If a < 0, as x → +∞, the function approaches negative infinity, and as x → -∞, the function approaches positive infinity.

   Examples:

   • For f(x) = x^3, as x becomes very large (positive or negative), f(x) becomes very large and maintains the same sign.
   • For f(x) = -5x^5, as x becomes very large (positive or negative), f(x) becomes very large and switches sign.

3. When n is Not an Integer:

   • The behavior depends on the value of n and the sign of a. If a > 0, the end behavior will be similar to an even or odd integer exponent, depending on whether n is greater or less than 1. If a < 0, the end behavior will be the opposite of that for a > 0.

   Examples:

   • For f(x) = 2x^0.5, as x → +∞, the function approaches positive infinity, and as x → -∞, the function approaches negative infinity.
   • For f(x) = -4x^1.5, as x → +∞, the function approaches negative infinity, and as x → -∞, the function approaches positive infinity.

In summary, the end behavior of a power function is determined by the values of the coefficient a (positive or negative) and the exponent n (even or odd). Understanding this behavior is crucial for analyzing the long-term trends of power functions.

Given the power function $�(�)=�{�}^{�}$ where $�$ is a non-negative integer, we can identify the end behavior based on the values of $�$ and $�$. Here are the general rules for the end behavior:

1. When $�$ is an Even Integer:

   • If $�>0$, as $�\to +\mathrm{\infty }$, the function approaches positive infinity, and as $�\to -\mathrm{\infty }$, the function approaches positive infinity.
   • If $�<0$, as $�\to +\mathrm{\infty }$, the function approaches negative infinity, and as $�\to -\mathrm{\infty }$, the function approaches negative infinity.

   Examples:

   • For $�(�)=2{�}^2$, as $�$ becomes very large (positive or negative), $�(�)$ becomes very large and positive.
   • For $�(�)=-3{�}^4$, as $�$ becomes very large (positive or negative), $�(�)$ becomes very large and negative.

2. When $�$ is an Odd Integer:

   • If $�>0$, as $�\to +\mathrm{\infty }$, the function approaches positive infinity, and as $�\to -\mathrm{\infty }$, the function approaches negative infinity.
   • If $�<0$, as $�\to +\mathrm{\infty }$, the function approaches negative infinity, and as $�\to -\mathrm{\infty }$, the function approaches positive infinity.

   Examples:

   • For $�(�)={�}^3$, as $�$ becomes very large (positive or negative), $�(�)$ becomes very large and maintains the same sign.
   • For $�(�)=-5{�}^5$, as $�$ becomes very large (positive or negative), $�(�)$ becomes very large and switches sign.

3. When $�$ is Not an Integer (Fractional or Decimal):

   • The behavior depends on the value of $�$ and the sign of $�$. If $�>0$, the end behavior will be similar to an even or odd integer exponent, depending on whether $�$ is greater or less than 1. If $�<0$, the end behavior will be the opposite of that for $�>0$.

   Examples:

   • For $�(�)=2{�}^{0.5}$, as $�\to +\mathrm{\infty }$, the function approaches positive infinity, and as $�\to -\mathrm{\infty }$, the function approaches negative infinity.
   • For $�(�)=-4{�}^{1.5}$, as $�\to +\mathrm{\infty }$, the function approaches negative infinity, and as $�\to -\mathrm{\infty }$, the function approaches positive infinity.

In the case of a non-negative integer $�$, you can determine whether $�$ is even or odd and follow the corresponding rules for even or odd exponents. The sign of $�$ also plays a role in determining whether the function approaches positive or negative infinity.

Polynomial functions are mathematical functions that are defined by equations of the form:

$�(�)={�}_{�}{�}^{�}+{�}_{�-1}{�}^{�-1}+\dots +{�}_2{�}^2+{�}_1�+{�}_0$

Where:

• $�(�)$ is the dependent variable or the output of the function.
• $�$ is the independent variable or the input of the function.
• ${�}_{�},{�}_{�-1},\dots ,{�}_2,{�}_1,{�}_0$ are constants called coefficients.
• $�$ is a non-negative integer called the degree of the polynomial, and it represents the highest power of $�$ in the function.

To identify a polynomial function, you should look for the following characteristics:

1. Single Term: Polynomial functions consist of one or more terms, each of which is a product of a coefficient and a power of $�$. There are no radical expressions, trigonometric functions, or other non-polynomial functions within the equation.

2. Whole Number Exponents: The exponents of $�$ in each term must be non-negative integers. This means that $�$ is raised to whole number powers, and there are no fractional or negative exponents.

3. Addition and Subtraction: Terms in a polynomial function are combined using addition and subtraction operations. There are no other operations like multiplication, division, or roots within the terms.

4. Degree of the Polynomial: The degree of the polynomial is determined by the highest power of $�$ among all the terms. For example:

   • If the highest power of $�$ is 2, it's a quadratic polynomial (degree 2).
   • If the highest power of $�$ is 3, it's a cubic polynomial (degree 3).
   • If the highest power of $�$ is $�$, it's an $�$-th degree polynomial.

5. Coefficients: The coefficients ${�}_{�},{�}_{�-1},\dots ,{�}_2,{�}_1,{�}_0$ can be any real numbers, including zero. They determine the specific values and shape of the polynomial.

Here are some examples of polynomial functions:

• $�(�)=3{�}^4-2{�}^3+5{�}^2-�+7$ (a fourth-degree polynomial).
• $�(�)=2{�}^2+4�-1$ (a quadratic polynomial).
• $ℎ(�)=5{�}^3-3�$ (a cubic polynomial).

These examples meet all the criteria mentioned above and are considered polynomial functions. Polynomial functions are widely used in mathematics and science for various applications, including curve fitting, modeling data, and solving equations.

To identify the degree and leading coefficient of a polynomial function, you should examine the expression carefully and apply the following rules:

1. Degree of the Polynomial:

   • The degree of a polynomial is the highest power of the variable (usually $�$) in the polynomial expression.
   • To find the degree, look for the term with the highest exponent of $�$ and determine its power.

2. Leading Coefficient:

   • The leading coefficient is the coefficient of the term with the highest power of the variable.
   • It is the constant that multiplies the highest power of $�$.

Here are some examples to illustrate how to identify the degree and leading coefficient of polynomial functions:

Example 1: Identify the degree and leading coefficient of the polynomial function $�(�)=3{�}^4-2{�}^3+5{�}^2-�+7$.

• Degree: The highest power of $�$ is ${�}^4$, so the degree of the polynomial is 4.

• Leading Coefficient: The coefficient of the term with the highest power of $�$ (${�}^4$) is 3. Therefore, the leading coefficient is 3.

Example 2: Identify the degree and leading coefficient of the polynomial function $�(�)=2{�}^2+4�-1$.

• Degree: The highest power of $�$ is ${�}^2$, so the degree of the polynomial is 2.

• Leading Coefficient: The coefficient of the term with the highest power of $�$ (${�}^2$) is 2. Therefore, the leading coefficient is 2.

Example 3: Identify the degree and leading coefficient of the polynomial function $ℎ(�)=5{�}^3-3�$.

• Degree: The highest power of $�$ is ${�}^3$, so the degree of the polynomial is 3.

• Leading Coefficient: The coefficient of the term with the highest power of $�$ (${�}^3$) is 5. Therefore, the leading coefficient is 5.

In summary, to identify the degree of a polynomial function, find the highest power of the variable (usually $�$), and to identify the leading coefficient, determine the coefficient of the term with that highest power. These characteristics are important for understanding the behavior and properties of polynomial functions.

The end behavior of a polynomial function refers to how the function behaves as the input variable $�$ approaches positive infinity ($+\mathrm{\infty }$) and negative infinity ($-\mathrm{\infty }$). Understanding the end behavior is crucial for characterizing the long-term behavior of a polynomial function and determining whether it rises or falls without bound.

To identify the end behavior of a polynomial function, you need to consider the degree of the polynomial and the leading coefficient. Here are the rules for identifying the end behavior:

1. Degree of the Polynomial:

   • The degree of a polynomial is determined by the highest power of $�$ in the polynomial expression.
   • If the degree is even (e.g., 2, 4, 6, etc.), the end behavior is the same on both sides of the y-axis.
   • If the degree is odd (e.g., 1, 3, 5, etc.), the end behavior is different on opposite sides of the y-axis.

2. Leading Coefficient:

   • The leading coefficient is the coefficient of the term with the highest power of $�$. It's the constant that multiplies the highest power of $�$.

Now, let's determine the end behavior based on the degree and leading coefficient:

Even-Degree Polynomial (Degree is Even):

• If the leading coefficient is positive ($�>0$), the polynomial rises on both sides of the y-axis as $�$ goes to $-\mathrm{\infty }$ and $+\mathrm{\infty }$.
• If the leading coefficient is negative ($�<0$), the polynomial falls on both sides of the y-axis as $�$ goes to $-\mathrm{\infty }$ and $+\mathrm{\infty }$.

Odd-Degree Polynomial (Degree is Odd):

• If the leading coefficient is positive ($�>0$), the polynomial rises to the right ($+\mathrm{\infty }$) and falls to the left ($-\mathrm{\infty }$) of the y-axis.
• If the leading coefficient is negative ($�<0$), the polynomial falls to the right ($+\mathrm{\infty }$) and rises to the left ($-\mathrm{\infty }$) of the y-axis.

Here are examples to illustrate the end behavior of polynomial functions:

1. For the polynomial $�(�)=2{�}^4-3{�}^3+{�}^2-4$ (even-degree and positive leading coefficient), as $�$ goes to $-\mathrm{\infty }$ and $+\mathrm{\infty }$, the function rises on both sides.

2. For the polynomial $�(�)=-{�}^3+4{�}^2-2�+1$ (odd-degree and negative leading coefficient), as $�$ goes to $-\mathrm{\infty }$ and $+\mathrm{\infty }$, the function falls on one side and rises on the other side.

Understanding the end behavior helps you visualize how the polynomial behaves as $�$ becomes very large (positive or negative). It's an important concept in analyzing and graphing polynomial functions.

To identify the local behavior of a polynomial function, you need to examine the function's behavior near specific points on the graph, specifically around its critical points and inflection points. Local behavior refers to how the function behaves in the vicinity of these points.

Here are the key steps to identify the local behavior of a polynomial function:

1. Determine the Degree and Leading Coefficient:

   • First, determine the degree of the polynomial by finding the highest power of the variable (usually $�$) in the polynomial expression.
   • Identify the leading coefficient, which is the coefficient of the term with the highest power of the variable.

2. Locate Critical Points:

   • Critical points are the values of $�$ where the derivative of the function is either zero or undefined.
   • To find critical points, calculate the first derivative of the polynomial function (${�}^{\mathrm{\prime }}(�)$), set it equal to zero, and solve for $�$.
   • Critical points are potential points where the function may have local extrema (maxima or minima).

3. Analyze Behavior at Critical Points:

   • To determine the local behavior at critical points, use the first and second derivative tests.
   • Evaluate the sign of the first derivative (${�}^{\mathrm{\prime }}(�)$) to determine whether the function is increasing or decreasing on intervals around the critical points.
   • Use the second derivative (${�}^{\mathrm{\prime }\mathrm{\prime }}(�)$) to determine concavity (whether the function is concave up or down).
   • The combination of increasing/decreasing and concave up/down helps identify local maxima, minima, and inflection points.

4. Identify Inflection Points (if applicable):

   • Inflection points are points on the graph where the concavity changes.
   • To find inflection points, set the second derivative (${�}^{\mathrm{\prime }\mathrm{\prime }}(�)$) equal to zero and solve for $�$.
   • Analyze the behavior of the function near these points to determine whether they are inflection points.

5. Sketch a Rough Graph:

   • Based on your analysis of the local behavior at critical points and inflection points, sketch a rough graph of the polynomial function to visualize its behavior.

6. Check End Behavior:

   • If needed, determine the end behavior of the polynomial function as $�$ approaches $-\mathrm{\infty }$ and $+\mathrm{\infty }$ to understand its behavior at extreme values of $�$.

7. Verify with Technology (optional):

   • You can use graphing calculators or graphing software to verify your analysis and get a more accurate representation of the graph.

By following these steps, you can identify and describe the local behavior of a polynomial function, including the presence of local extrema, inflection points, and the shape of the graph in different regions. This analysis is important for understanding the function's behavior in detail.

In the context of polynomial functions, "intercepts" typically refer to x-intercepts and y-intercepts, while "turning points" refer to points on the graph where the function changes direction. Here's how to find and interpret both intercepts and turning points of polynomial functions:

1. X-Intercepts (Zeros or Roots):

   • X-intercepts are the points where the graph of the polynomial function intersects the x-axis.
   • To find the x-intercepts, set $�(�)=0$ and solve for $�$. The solutions to this equation are the x-intercepts.
   • Each x-intercept corresponds to a real root or zero of the polynomial. If a root occurs with multiplicity greater than one, it represents a repeated x-intercept.

   For example, if $�(�)={�}^2-4$, to find the x-intercepts: ${�}^2-4=0$ $(�-2)(�+2)=0$ $�=2\text{ and }�=-2$ So, the x-intercepts are $�=2$ and $�=-2$.

2. Y-Intercept:

   • The y-intercept is the point where the graph of the polynomial function intersects the y-axis.
   • To find the y-intercept, evaluate the function at $�=0$, which gives you the value of $�(0)$. The point $(0,�(0))$ is the y-intercept.

   For example, if $�(�)=2{�}^3-3�+1$, to find the y-intercept: $�(0)=2(0{)}^3-3(0)+1=1$ So, the y-intercept is at $(0,1)$.

3. Turning Points (Local Extrema):

   • Turning points are points on the graph where the function changes direction, going from increasing to decreasing or vice versa.
   • Turning points correspond to local extrema, which can be maxima or minima.
   • To find turning points, you need to locate critical points by setting the first derivative ${�}^{\mathrm{\prime }}(�)$ equal to zero and solving for $�$. These critical points represent potential turning points.
   • Use the second derivative test to determine whether each critical point is a local maximum, local minimum, or neither.

   For example, if $�(�)={�}^3-6{�}^2+9�+2$:

   • Find critical points by solving ${�}^{\mathrm{\prime }}(�)=0$.
   • Use the second derivative test to determine whether these points are maxima, minima, or neither.

4. Inflection Points:

   • Inflection points are points on the graph where the function changes concavity (from concave up to concave down or vice versa).
   • To find inflection points, locate the points where the second derivative ${�}^{\mathrm{\prime }\mathrm{\prime }}(�)$ is equal to zero, and then use the concavity test to determine whether they are inflection points.

   For example, if $�(�)=3{�}^4-4{�}^3+2{�}^2-7$:

   • Find points where ${�}^{\mathrm{\prime }\mathrm{\prime }}(�)=0$.
   • Use the concavity test to determine whether these points are inflection points.

Understanding x-intercepts, y-intercepts, turning points, and inflection points helps you gain insight into the behavior of polynomial functions and how they interact with the coordinate axes and change in direction or curvature on their graphs.

Smooth and continuous graphs are both characteristics of functions and relations represented on a coordinate plane, but they refer to slightly different aspects of the graph. Here's a comparison of the two:

1. Smooth Graphs:

   • A smooth graph typically refers to a graph that has no sharp corners, jumps, or discontinuities. It appears visually smooth and without abrupt changes in direction.
   • In mathematical terms, a smooth graph corresponds to a function or relation that has continuous derivatives at all points in its domain.
   • Smooth graphs are often associated with functions that are differentiable (i.e., they have derivatives) and have no vertical asymptotes or holes in the graph.
   • Examples of smooth graphs include the graphs of most polynomial functions, trigonometric functions, and exponential functions.

2. Continuous Graphs:

   • A continuous graph refers to a graph that is unbroken or without interruptions over its entire domain. There are no "jumps" or "holes" in the graph.
   • In mathematical terms, a continuous graph corresponds to a function or relation where there are no gaps or jumps in the values of the function. It may have vertical asymptotes, but it is still considered continuous.
   • Continuous graphs may have smooth or non-smooth sections, but the key is that they are not interrupted or discontinuous.
   • Examples of continuous graphs include the graphs of piecewise functions, rational functions (except where they have vertical asymptotes), and functions like the absolute value function.

In summary, the terms "smooth" and "continuous" are related but not synonymous:

• A smooth graph generally implies that the graph is visually smooth, without sharp corners or abrupt changes in direction, often associated with differentiable functions.
• A continuous graph implies that there are no gaps, jumps, or interruptions in the graph over its entire domain, even if it has vertical asymptotes or other features.

It's important to consider both smoothness and continuity when analyzing and interpreting graphs in mathematics and science.

Here are examples of graphs that illustrate the concepts of smooth and continuous graphs:

1. Smooth and Continuous Graph:

   • The graph of the function $�(�)={�}^2$ is both smooth and continuous over its entire domain. It is a parabola that has no jumps, holes, or sharp corners.

   
   

2. Smooth but Not Continuous:

   • The graph of the piecewise function $�(�)=\{\begin{array}{ll}{�}^2,& �\ge 0\\ -{�}^2,& �<0\end{array}$ is smooth on each branch but not continuous at $�=0$. There's a jump or discontinuity at $�=0$.

   

3. Continuous but Not Smooth:

   • The graph of the absolute value function $�(�)=\mathrm{\mid }�\mathrm{\mid }$ is continuous but not smooth at $�=0$. It has a sharp corner at the origin.

   

4. Neither Smooth nor Continuous:

   • The graph of the step function $�(�)=\{\begin{array}{ll}1,& �>0\\ 0,& �\le 0\end{array}$ is neither smooth nor continuous. It has a step or jump discontinuity at $�=0$.

   

These examples illustrate the differences between smooth and continuous graphs. Smoothness relates to the absence of sharp changes in direction, while continuity relates to the absence of gaps, jumps, or interruptions in the graph. A graph can exhibit one or both of these characteristics, depending on the function or relation it represents.