Piecewise Functions

Definition and Graphing

Piecewise functions are defined by different expressions depending on the input value. Each 'piece' applies to a specific interval of the domain.

• Piecewise Function: A function composed of multiple sub-functions, each with its own domain.

• Graphing: Plot each piece on its respective interval, ensuring endpoints are correctly marked (open or closed circles).

• Example: Given: Graph each segment on its specified interval.

Evaluating Piecewise Functions

To evaluate a piecewise function at a specific value, determine which interval the input falls into and use the corresponding formula.

• Example: For as above, find and by substituting into the correct piece.

Function Transformations

Types of Transformations

Transformations alter the graph of a function by shifting, stretching, compressing, or reflecting it.

• Vertical Shifts: shifts the graph up/down.

• Horizontal Shifts: shifts the graph right/left.

• Reflections: reflects over the x-axis; reflects over the y-axis.

• Stretch/Compression: stretches () or compresses () vertically.

• Example: is a square root function shifted left by 5 units.

Quadratic Functions

Standard and Vertex Form

Quadratic functions are polynomials of degree 2. Their graphs are parabolas.

• Standard Form:

• Vertex Form: , where is the vertex.

• Axis of Symmetry: in vertex form, or in standard form.

• Domain: All real numbers,

• Range: Depends on ; if , range is ; if , range is

• Example:

Finding the Equation from a Graph

Given a graph, identify the vertex and another point to write the equation in standard form.

• Example: If the vertex is at and another point is , use vertex form and expand.

Polynomial Functions

Degree, Leading Coefficient, and Intercepts

Polynomial functions are sums of terms with non-negative integer exponents.

• Degree: The highest exponent of .

• Leading Coefficient (L.C.): The coefficient of the term with the highest degree.

• x-intercepts: Values of where ; multiplicity indicates how the graph behaves at the intercept.

• y-intercept: Value of .

• End Behavior: Determined by degree and leading coefficient.

• Example:

Multiplicity

• Multiplicity: If is a factor, is a root of multiplicity .

• If is even, the graph touches and turns at ; if odd, it crosses the axis.

Rational Functions

Vertical and Horizontal Asymptotes

Rational functions are quotients of polynomials. Their graphs may have asymptotes and holes.

• Vertical Asymptotes (V.A.): Values of where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes (H.A.): Determined by degrees of numerator and denominator:

  • If degree numerator < degree denominator:

  • If degrees equal:

  • If degree numerator > degree denominator: No H.A.; may have slant asymptote.

• Slant Asymptote (S.A.): If degree numerator is exactly one more than denominator, divide to find S.A.

• Example:

Intercepts and Holes

• x-intercepts: Set numerator equal to zero (excluding values that make denominator zero).

• y-intercept: Evaluate .

• Holes: Occur at values where both numerator and denominator are zero (common factors).

Summary Table: Asymptotes and Intercepts

Square Root and Absolute Value Functions

Graphing and Transformations

Square root and absolute value functions have distinctive graphs and respond to transformations.

• Square Root Function: ; domain .

• Absolute Value Function: ; V-shaped graph.

• Transformations: Apply shifts, stretches, and reflections as with other functions.

• Example: is an absolute value function shifted right by 7, stretched by , and down by 4.

Summary of Key Concepts

• Piecewise functions require careful attention to intervals.

• Transformations affect the position and shape of graphs.

• Quadratic and polynomial functions are analyzed via degree, leading coefficient, intercepts, and end behavior.

• Rational functions involve asymptotes, intercepts, and holes.

• Square root and absolute value functions have restricted domains and unique graph shapes.

Additional info: The study notes expand on brief worksheet questions, providing definitions, formulas, and academic context for Precalculus topics such as piecewise functions, transformations, quadratics, polynomials, rational functions, and graphing techniques.