Linear and Quadratic Functions and Modeling 2.1 Linear and Quadratic Functions and Modeling

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What you’ll learn about Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Linear Correlation and Modeling Quadratic Functions and Their Graphs Applications of Quadratic Functions … and why Many business and economic problems are modeled by linear functions. Quadratic and higher degree polynomial functions are used to model some manufacturing applications.

Polynomial Function

Polynomial Functions of No and Low Degree Name Form Degree Zero Function f(x)=0 Undefined Constant Function f(x)=a (a≠0) 0 Linear Function f(x)=ax+b (a≠0) 1 Quadratic Function f(x)=ax2+bx+c (a≠0) 2

Example Finding an Equation of a Linear Function

Example Finding an Equation of a Linear Function

Average Rate of Change

Constant Rate of Change Theorem A function defined on all real numbers is a linear function if and only if it has a constant nonzero average rate of change between any two points on its graph.

Characterizing the Nature of a Linear Function Point of View Characterization Verbal polynomial of degree 1 Algebraic f(x) = mx + b (m≠0) Graphical slant line with slope m and y-intercept b Analytical function with constant nonzero rate of change m: f is increasing if m>0, decreasing if m<0; initial value of the function = f(0) = b

Properties of the Correlation Coefficient, r When r > 0, there is a positive linear correlation. When r < 0, there is a negative linear correlation. When |r| ≈ 1, there is a strong linear correlation. When |r| ≈ 0, there is weak or no linear correlation.

Linear Correlation

Regression Analysis Enter and plot the data (scatter plot). Find the regression model that fits the problem situation. Superimpose the graph of the regression model on the scatter plot, and observe the fit. Use the regression model to make the predictions called for in the problem.

Example Transforming the Squaring Function

Example Transforming the Squaring Function

The Graph of f(x)=ax2

Vertex Form of a Quadratic Equation Any quadratic function f(x) = ax2 + bx + c, a≠0, can be written in the vertex form f(x) = a(x – h)2 + k The graph of f is a parabola with vertex (h,k) and axis x = h, where h = -b/(2a) and k = c – ah2. If a>0, the parabola opens upward, and if a<0, it opens downward.

Example Finding the Vertex and Axis of a Quadratic Function

Example Finding the Vertex and Axis of a Quadratic Function

Characterizing the Nature of a Quadratic Function Point of View Characterization

Vertical Free-Fall Motion

Power Functions and Modeling 2.2 Power Functions and Modeling

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What you’ll learn about Power Functions and Variation Monomial Functions and Their Graphs Graphs of Power Functions Modeling with Power Functions … and why Power functions specify the proportional relationships of geometry, chemistry, and physics.

Power Function Any function that can be written in the form f(x) = k·xa, where k and a are nonzero constants, is a power function. The constant a is the power, and the k is the constant of variation, or constant of proportion. We say f(x) varies as the ath power of x, or f(x) is proportional to the ath power of x.

Example Analyzing Power Functions

Example Analyzing Power Functions

Monomial Function Any function that can be written as f(x) = k or f(x) = k·xn, where k is a constant and n is a positive integer, is a monomial function.

Example Graphing Monomial Functions

Example Graphing Monomial Functions

Graphs of Power Functions For any power function f(x) = k·xa, one of the following three things happens when x < 0. f is undefined for x < 0. f is an even function. f is an odd function.

Graphs of Power Functions

Polynomial Functions of Higher Degree with Modeling 2.3 Polynomial Functions of Higher Degree with Modeling

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What you’ll learn about Graphs of Polynomial Functions End Behavior of Polynomial Functions Zeros of Polynomial Functions Intermediate Value Theorem Modeling … and why These topics are important in modeling and can be used to provide approximations to more complicated functions, as you will see if you study calculus.

The Vocabulary of Polynomials

Example Graphing Transformations of Monomial Functions

Example Graphing Transformations of Monomial Functions

Cubic Functions

Quartic Function

Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.

Leading Term Test for Polynomial End Behavior

Example Applying Polynomial Theory

Example Applying Polynomial Theory

Example Finding the Zeros of a Polynomial Function

Example Finding the Zeros of a Polynomial Function

Multiplicity of a Zero of a Polynomial Function

Example Sketching the Graph of a Factored Polynomial

Example Sketching the Graph of a Factored Polynomial

Intermediate Value Theorem If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0=f(c) for some number c in [a,b].

Real Zeros of Polynomial Functions 2.4 Real Zeros of Polynomial Functions

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What you’ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower Bounds … and why These topics help identify and locate the real zeros of polynomial functions.

Division Algorithm for Polynomials

Example Using Polynomial Long Division

Example Using Polynomial Long Division

Remainder Theorem

Example Using the Remainder Theorem

Example Using the Remainder Theorem

Factor Theorem

Example Using Synthetic Division

Example Using Synthetic Division

Rational Zeros Theorem

Upper and Lower Bound Tests for Real Zeros

Example Finding the Real Zeros of a Polynomial Function

Example Finding the Real Zeros of a Polynomial Function

Example Finding the Real Zeros of a Polynomial Function

Example Finding the Real Zeros of a Polynomial Function

Complex Zeros and the Fundamental Theorem of Algebra 2.5 Complex Zeros and the Fundamental Theorem of Algebra

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What you’ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients … and why These topics provide the complete story about the zeros and factors of polynomials with real number coefficients.

Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

Linear Factorization Theorem

Fundamental Polynomial Connections in the Complex Case The following statements about a polynomial function f are equivalent if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x).

Example Exploring Fundamental Polynomial Connections

Example Exploring Fundamental Polynomial Connections

Complex Conjugate Zeros

Example Finding a Polynomial from Given Zeros

Example Finding a Polynomial from Given Zeros

Factors of a Polynomial with Real Coefficients Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.

Example Factoring a Polynomial

Example Factoring a Polynomial

Graphs of Rational Functions 2.6 Graphs of Rational Functions

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What you’ll learn about Rational Functions Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions … and why Rational functions are used in calculus and in scientific applications such as inverse proportions.

Rational Functions

Example Finding the Domain of a Rational Function

Example Finding the Domain of a Rational Function

Graph a Rational Function

Graph a Rational Function

Example Finding Asymptotes of Rational Functions

Example Finding Asymptotes of Rational Functions

Example Graphing a Rational Function

Example Graphing a Rational Function

Solving Equations in One Variable 2.7 Solving Equations in One Variable

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What you’ll learn about Solving Rational Equations Extraneous Solutions Applications … and why Applications involving rational functions as models often require that an equation involving fractions be solved.

Extraneous Solutions When we multiply or divide an equation by an expression containing variables, the resulting equation may have solutions that are not solutions of the original equation. These are extraneous solutions. For this reason we must check each solution of the resulting equation in the original equation.

Example Solving by Clearing Fractions

Example Solving by Clearing Fractions

Example Eliminating Extraneous Solutions

Example Eliminating Extraneous Solutions

Example Finding a Minimum Perimeter

Example Finding a Minimum Perimeter

Solving Inequalities in One Variable 2.8 Solving Inequalities in One Variable

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What you’ll learn about Polynomial Inequalities Rational Inequalities Other Inequalities Applications … and why Designing containers as well as other types of applications often require that an inequality be solved.

Polynomial Inequalities

Example Finding where a Polynomial is Zero, Positive, or Negative

Example Finding where a Polynomial is Zero, Positive, or Negative -3 4 (-)(-)2 (+)(-)2 (+)(+)2 negative positive

Example Solving a Polynomial Inequality Graphically

Example Solving a Polynomial Inequality Graphically

Example Creating a Sign Chart for a Rational Function

Example Creating a Sign Chart for a Rational Function -3 1 (-) (-)(-) negative positive -1 (+)(-) (+) (+)(+) und.

Example Solving an Inequality Involving a Radical

Example Solving an Inequality Involving a Radical -1 2 (-)(+) (+)(+) undefined positive negative

Chapter Test

Chapter Test

Chapter Test

Chapter Test Solutions

Chapter Test Solutions

Chapter Test Solutions