Functions: Transformations, Combinations, and Inverses

Transformations of Functions

Transformations allow us to modify the graph of a function in systematic ways. Understanding these helps in graphing and analyzing functions efficiently.

• Vertical and Horizontal Shifts: Shifting a function up/down or left/right without changing its shape.

  • Vertical shift: shifts the graph up by units if , down if .

  • Horizontal shift: shifts the graph right by units if , left if .

• Vertical and Horizontal Dilations (Stretching/Shrinking): Changing the steepness or width of the graph.

  • Vertical dilation: stretches the graph vertically by if , compresses if .

  • Horizontal dilation: compresses the graph horizontally by if , stretches if .

• Reflections: Flipping the graph over an axis.

  • Reflection over the x-axis: .

  • Reflection over the y-axis: .

• Graphs of Common Functions: Recognizing basic shapes such as lines, quadratics, square roots, cubics, etc.

Example: The graph of is obtained by shifting left by 3, reflecting over the x-axis, stretching vertically by 2, and shifting up by 1.

Combinations and Compositions of Functions

Functions can be combined or composed to create new functions, which is essential for modeling complex relationships.

• Domains of Functions: The set of all input values () for which the function is defined.

• Combining Functions: Given and :

  • Addition:

  • Subtraction:

  • Multiplication:

  • Division: ,

• Composition of Functions: means apply first, then .

Example: If and , then .

Inverse of Functions

The inverse of a function reverses the effect of the original function. Not all functions have inverses; only one-to-one functions do.

• Definition: The inverse satisfies and .

• One-to-One Functions: A function is one-to-one if each output is produced by exactly one input (passes the horizontal line test).

• Verifying Inverses: Check if and .

• Finding Inverses: To find the inverse, solve for , then swap and .

• Graphing Inverses: The graph of is the reflection of over the line .

Example: For , solve for in terms of to find the inverse.

Quadratic Functions

End Behavior of Quadratic Functions

Quadratic functions have the general form . The end behavior depends on the sign of .

• Upward Opening ("Cupping Up"): If , the parabola opens upward.

• Downward Opening ("Cupping Down"): If , the parabola opens downward.

Important Points of Quadratics

• x-intercepts: Points where . Solve .

• y-intercept: Point where , so .

• Vertex: The turning point of the parabola.

  • Vertex formula:

  • Vertex -value:

Example: For , the vertex is at , .

Graphing Quadratic Functions

• Plot the vertex, axis of symmetry (), and intercepts.

• Determine end behavior from the sign of .

• Draw the parabola accordingly.

Maxima and Minima of Quadratic Functions

• If , the vertex is a minimum point.

• If , the vertex is a maximum point.

Example: For , the maximum occurs at the vertex.

Applications of Quadratic Functions

• Quadratic functions model projectile motion, area optimization, and other real-world scenarios.

• Optimization problems often involve finding the maximum or minimum value of a quadratic function.

Example: Maximizing the area of a rectangle with a fixed perimeter using a quadratic function.

Challenging Problems: Sample Applications

Transformation Problem

• Given: (parent function).

• Task: Find the equation for after a series of transformations (e.g., shifts, reflections, dilations).

• Approach: Identify each transformation from the graph and write the new equation accordingly.

Function Composition in Context

• Given: (subtracts $400g(x) = 0.75x$ (applies a 25% discount).

• Task: Find and and interpret which is a better deal.

• Compositions:

• Interpretation: gives a higher final price, so is a better deal for the consumer.

Finding the Inverse of a Rational Function

• Given:

• Task: Find .

• Solution:

  • Let

  • Solve for :

  • Swap and for the inverse:

Quadratic Application: Area Optimization

• Given: 700 feet of fencing to enclose a rectangle next to a river (one side unfenced).

• Let: = length of the short side (perpendicular to river).

• Perimeter constraint: (since only three sides are fenced).

• Area function:

• Express in terms of :

• So,

• Maximum Area: Since the quadratic opens downward (), there is a maximum.

• Vertex:

• Dimensions for maximum area: ,

• Maximum area: square feet

Summary Table: Key Properties of Quadratic Functions

Study Tips for Success

• Start with your weakest topics and practice problems from those areas first.

• Create and take practice exams under timed conditions to simulate the test environment.

• Study in groups to benefit from different perspectives and strengths.

• Utilize academic support centers and office hours for additional help.

• Study consistently, but allow time for rest before the exam to consolidate learning.

Additional info: The above notes synthesize the review sheet's outline and expand on each topic with definitions, formulas, and examples for comprehensive exam preparation.