College Algebra: Polynomial and Rational Functions, Optimization, and Systems of Equations

Polynomial Functions

Vertex Form and Maximum/Minimum Values

Polynomial functions, especially quadratics, can be written in vertex form to easily identify their maximum or minimum values. The vertex form is useful for graphing and solving optimization problems.

• Vertex Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

• Finding the Vertex: For $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$.

• Maximum/Minimum: If $a < 0$, the parabola opens downward (maximum); if $a > 0$, it opens upward (minimum).

• Example: For $f(x) = -2x^2 + 4x + 1$, vertex at $x = -\frac{4}{2(-2)} = 1$.

Optimization Problem Example

Optimization problems often involve maximizing or minimizing an area or volume given certain constraints. A classic example is maximizing the area of a rectangle adjacent to a river with a fixed amount of fencing.

• Problem: Maximize area $A$ of a rectangle with one side along a river, using 800 ft of fencing for the other three sides.

• Let: $x$ = width (perpendicular to river), $y$ = length (parallel to river).

• Constraint: $2x + y = 800$

• Area: $A = x \cdot y$

• Substitute: $y = 800 - 2x \implies A(x) = x(800 - 2x) = 800x - 2x^2$

• Find Maximum: Vertex at $x = -\frac{800}{2 \times -2} = 200$

• Dimensions: $x = 200$ ft, $y = 800 - 2 \times 200 = 400$ ft

• Maximum Area: $A = 200 \times 400 = 80,000$ ft$^2$

Polynomial and Rational Functions

Identifying Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, involving only non-negative integer exponents.

• Examples: $3x^2 + 2x + 1$ (polynomial), $\frac{1}{x}$ (not a polynomial)

• Degree: Highest exponent of the variable

End Behavior and Leading Coefficient Test

The end behavior of a polynomial function describes how the function behaves as $x \to \infty$ or $x \to -\infty$.

• Even Degree, Positive Leading Coefficient: Both ends up

• Even Degree, Negative Leading Coefficient: Both ends down

• Odd Degree, Positive Leading Coefficient: Left down, right up

• Odd Degree, Negative Leading Coefficient: Left up, right down

Multiplicity of Zeros

The multiplicity of a zero determines how the graph behaves at the x-intercept.

Multiplicity	Even	Odd
Graph Behavior	Bounces	Crosses through

Rational Functions

Asymptotes

Rational functions can have vertical, horizontal, or oblique (slant) asymptotes.

• Vertical Asymptotes: Set denominator equal to zero and solve for $x$.

• Horizontal Asymptotes: Compare degrees of numerator and denominator.

• Oblique Asymptotes: Occur when degree of numerator is one more than denominator.

Sketching Rational Functions

1. Find asymptotes

2. Determine intercepts

3. Find domain

4. Sketch the graph

Systems of Equations and Inequalities

Solving Systems of Equations

Systems of equations can be solved graphically or algebraically (substitution, elimination).

• Graphical Solution: Intersection point(s) of the graphs represent solutions.

• No Solution: Parallel lines

• Infinite Solutions: Coincident lines

Solving Systems of Inequalities

Systems of inequalities are solved by graphing the solution regions for each inequality and finding the overlap.

• Shaded Region: Represents all solutions that satisfy all inequalities in the system.

• Boundary Lines: Solid for $\leq$ or $\geq$, dashed for $<$ or $>$.

Summary of Key Concepts

• Graphing maximum and minimum parabolas

• Understanding end behavior and zeros of polynomials

• Finding zeros and multiplicities of polynomials

• Finding vertical and horizontal asymptotes

• Solving systems of equations and inequalities