Unit 1: Functions, Transformations & Concavity

Section 2.4: Function Shifts & Transformations

Function transformations are fundamental in precalculus, allowing us to manipulate and analyze the behavior of functions. Shifts move the graph horizontally or vertically, while other transformations can stretch, compress, or reflect the graph.

• Vertical Shifts: shifts the graph of up by units; shifts it down by $k$ units.

• Horizontal Shifts: shifts left by units; shifts it right by $k$ units.

• Point Transformation Rule: If is a point on , then on , the new point is .

• Example: If lies on , then on , the corresponding point is .

Section 2.5: Composition of Functions & Inverse Functions

Function composition and inverses are key concepts for understanding how functions interact and how to reverse their effects. The composition applies one function to the result of another, while the inverse undoes the action of the original function.

• Function Composition: . Apply first, then .

• Invertibility: A function has an inverse if and only if it passes the Horizontal Line Test (HLT).

• Inverse Property: and .

• Switching Table/Point Values: If , then .

• Finding an Inverse Formula:

  1. Replace with .

  2. Swap and .

  3. Solve for .

  4. Replace with .

Section 2.6: Concavity & Rates of Change

Concavity describes how a function bends, and rates of change measure how quickly a function's output changes. These concepts are important for analyzing graphs and understanding function behavior.

• Concave Up: The graph bends upward (like a cup, ). The average rate of change is increasing.

• Concave Down: The graph bends downward (like a frown, ). The average rate of change is decreasing.

• Concavity & Monotonicity:

  • Concave Up & Increasing: Graph rises and bends upward.

  • Concave Up & Decreasing: Graph falls but bends upward.

  • Concave Down & Increasing: Graph rises but bends downward.

  • Concave Down & Decreasing: Graph falls and bends downward.

Unit 2: Quadratic Functions

Section 3.1: Quadratic Functions & Word Problems

Quadratic functions are polynomials of degree two and are widely used in modeling real-world situations such as projectile motion and area maximization. Their graphs are parabolas, and their properties depend on the coefficients.

• Standard Form:

• Factored Form: , where and are the zeros.

• Vertex Location: ; -value is .

• Concavity:

  • If , parabola opens up (vertex is a minimum).

  • If , parabola opens down (vertex is a maximum).

• Applied Word Problems:

  • Projectile Motion: ; maximum height at the vertex; time when it hits the ground is the positive zero.

  • Area Maximization: Set up subject to constraints (e.g., perimeter).

Unit 3: Exponential Functions & Models

Section 4.1 & 4.2: Exponential Growth/Decay vs. Linear Models

Exponential functions model situations where quantities grow or decay by a constant percentage. Linear models change by a constant amount, while exponential models change by a constant ratio.

• Exponential Model:

• Parameters:

  • : Initial amount

  • : Growth/decay factor ()

  • : Growth/decay rate (decimal)

• Linear vs. Exponential:

  • Linear: Constant absolute change (constant slope).

  • Exponential: Constant percentage/ratio change.

• Finding Exponential Equation Given Two Points:

  1. Set up: ,

  2. Divide:

  3. Solve for , then substitute to find .

Section 4.3: Graphing Exponential Functions & Asymptotes

Exponential functions have characteristic graphs with horizontal asymptotes. Their end behavior depends on the base.

• Horizontal Asymptote: for .

• End Behavior:

  • Exponential Decay (): As , .

  • Exponential Growth (): As , .

Section 4.4: Compound Interest & APY

Compound interest is a financial application of exponential functions. APY (Annual Percentage Yield) measures the effective annual rate, accounting for compounding.

• Periodic Compound Interest:

  • Formula:

  • : Principal (initial deposit)

  • : Annual interest rate (decimal)

  • : Number of compounding periods per year (e.g., for annually, for monthly)

  • : Time in years

• Annual Percentage Yield (APY):

  • Formula:

  • APY represents the actual yearly percentage increase, including compounding effects.

Section 4.5: Continuous Growth/Decay & Base e

Continuous compounding uses the mathematical constant and models growth or decay that happens at every instant. The continuous rate relates to the annual factor via logarithms.

• Continuous Growth Formula: or

• Parameters:

  • or : Initial amount

  • : Continuous growth rate ( for growth, for decay)

• Conversion Between Annual Factor and Continuous Rate:

• Example Comparison:

  • Investment of $1000 years at APR compounded monthly:

  • Investment of $1000 years at compounded continuously:

Comprehensive Formula Summary Sheet