Unit 1: Functions, Transformations & Concavity (Sections 2.4 – 2.6)

Section 2.4: Function Shifts & Transformations

Understanding how functions change under various transformations is essential for graphing and modeling. Shifts and transformations allow us to predict and describe changes in function behavior.

• Vertical Shifts: shifts the graph of up by units. shifts the graph down by units.

• Horizontal Shifts: shifts the graph left by units. shifts the graph right by units.

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

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

Section 2.5: Composition of Functions & Inverse Functions

Composing functions and finding inverses are foundational skills for analyzing complex relationships between variables.

• Function Composition: . Apply first, then to the result.

• Invertibility & Horizontal Line Test: A function has an inverse if and only if it passes the Horizontal Line Test (each -value corresponds to exactly one -value).

• Inverse Property: and .

• Switching Table/Point Values: If , then .

• How to Find an Inverse Formula ():

  1. Replace with .

  2. Swap and .

  3. Solve for .

  4. Replace with .

Section 2.6: Concavity & Rates of Change

Concavity describes the direction a curve bends, while rates of change measure how quickly a function's output changes with respect to its input.

• 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.

• Visual Summary:

  • 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)

Standard and Factored Forms

Quadratic functions model parabolic relationships and are fundamental in algebra and applications such as projectile motion and area optimization.

• Standard Form:

• Factored Form: , where and are the zeros (-intercepts).

• Vertex Location: ; -coordinate is .

• Concavity:

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

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

Applied Word Problem Types

• Projectile Motion: (height as a function of time).

  • Maximum height at the vertex.

  • Time when it hits the ground: solve for .

• Area Maximization: Set up subject to a constraint (e.g., fixed perimeter). Example: Maximizing area with 80m of fencing along 3 sides of a river.

Unit 3: Exponential Functions & Models (Sections 4.1 – 4.5)

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

Exponential and linear models describe different types of change. Recognizing which model fits a situation is crucial for accurate predictions.

• Exponential Model:

  • : Initial amount

  • : Growth/decay factor ()

  • : Growth/decay rate (as a decimal)

• Linear vs. Exponential:

  • Linear: Constant absolute change (constant slope).

  • Exponential: Constant percentage/ratio change.

• Finding an Exponential Equation Given Two Points:

  1. Set up: and

  2. Divide:

  3. Solve for , then substitute to find .

Section 4.3: Graphing Exponential Functions & Asymptotes

• Horizontal Asymptote: All base exponential functions have a horizontal asymptote at .

• End Behavior:

  • Exponential Decay (): As , .

  • Exponential Growth (): As , .

Section 4.4: Compound Interest & APY

• Periodic Compound Interest:

  • : Principal (initial deposit)

  • : Annual interest rate (as a decimal)

  • : Number of compounding periods per year (e.g., 1, 4, 12, 365)

  • : Time in years

• Annual Percentage Yield (APY):

  • APY is the effective annual rate, accounting for compounding.

Section 4.5: Continuous Growth/Decay & Base

• Continuous Growth Formula: or

  • or : Initial amount

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

• Converting 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