Quadratic Functions

Definition and Properties

Quadratic functions are polynomial functions of degree 2, typically written as , where . They graph as parabolas and have important features such as vertex, axis of symmetry, and direction of opening.

• Vertex: The point where the parabola reaches its maximum or minimum. For , the vertex is at .

• Axis of Symmetry: The vertical line .

• Direction: If , the parabola opens upward; if , it opens downward.

• Intercepts: The y-intercept is . The x-intercepts are found by solving .

Example: For :

• Vertex: , ; vertex is .

• Opens downward since .

• Y-intercept: .

• X-intercepts: Solve .

Polynomial Functions from Graphs

Constructing Polynomial Equations

Polynomials can be constructed from given zeros and their multiplicities, as well as other graph features such as degree and intercepts.

• Zero: A value where .

• Multiplicity: The number of times a zero is repeated. If is a zero of multiplicity , then is a factor.

• Degree: The highest power of in the polynomial.

• Y-intercept: The value of the polynomial at .

Example: Zeros at (multiplicity 2), (multiplicity 1), (multiplicity 1), y-intercept , degree 4.

• General form:

• Find using the y-intercept:

Revenue Optimization

Maximizing Revenue with Quadratic Models

Revenue functions often take the form , where is price. The maximum revenue occurs at the vertex of the parabola.

• Maximum Revenue: For , vertex at .

• Maximum Value: .

Example: The price that maximizes revenue is ; maximum revenue is $1200$.

Polynomial Analysis

End Behavior, Zeros, and Turning Points

Analyzing polynomials involves understanding their end behavior, zeros, multiplicities, and turning points.

• End Behavior: Determined by the leading term. For , expand to find the leading term.

• Zeros: Solve ; multiplicities affect how the graph touches or crosses the x-axis.

• Y-intercept: .

• Turning Points: Maximum number is degree minus one.

Example: For a degree 4 polynomial, maximum 3 turning points.

Inequality Solving

Solving Polynomial Inequalities

To solve inequalities such as , find critical numbers and test intervals.

• Critical Numbers: Values where numerator or denominator is zero.

• Sign Chart: Test intervals between critical numbers to determine where the inequality holds.

• Interval Notation: Express solution as intervals where the inequality is true.

Example: Find zeros of numerator and denominator, plot on number line, test sign in each interval.

Polynomial Features

Degree, Zeros, and Holes

Polynomials may have multiple zeros, each with a specific multiplicity, and may exhibit holes if factors cancel in rational functions.

• Degree: The sum of the exponents in the expanded form.

• Zeros and Multiplicities: For , zeros at (multiplicity 2), (multiplicity 4), (multiplicity 1).

• Holes: Occur in rational functions when a factor cancels in numerator and denominator.

Graph Sketching

Sketching Polynomials from Features

To sketch a polynomial graph, use zeros, multiplicities, intercepts, and end behavior.

• Zeros: Mark on x-axis with multiplicity.

• Y-intercept: Mark on y-axis.

• End Behavior: For degree 3, leading coefficient determines direction.

Example: Zeros at (multiplicity 1), (multiplicity 2), y-intercept at , degree 3, end behavior like .

Complete Analysis of Rational Functions

Domain, Intercepts, Asymptotes, and Evaluation

Rational functions are quotients of polynomials. Analysis includes finding domain, intercepts, asymptotes, and evaluating at specific points.

• Domain: All real numbers except where denominator is zero.

• Intercepts: Set numerator or denominator to zero as appropriate.

• Asymptotes: Vertical at zeros of denominator; horizontal determined by degrees of numerator and denominator.

• Evaluation: Substitute values for .

Example: For :

• Domain:

• Vertical asymptotes at

• Horizontal asymptote at (degrees equal)

• Evaluate

Applied Problem: Maximizing Area

Optimization with Constraints

Applied problems often involve maximizing or minimizing quantities under constraints, such as fencing a garden.

• Function for Area: If one side is against a house, and is the width, total fencing is .

• Area:

• Express in terms of :

• Area as function of :

• Maximize : Find vertex of quadratic.

Example: Maximum area occurs at ; ; .

Study Tips

• Practice graphing functions by hand without a calculator.

• Memorize the vertex formula for quadratic functions.

• Understand how multiplicity affects graph behavior at zeros.

• Practice creating sign charts for inequalities.