Lesson 3.6b Graphing & Identifying Key Features of Exponential Functions Concept: Characteristics of Exponential Functions Lesson EQ: How do you graph, interpret, and apply the key features of an exponential function? (Standard F.IF.4,5,7) Vocabulary: Domain, Range, & End Behavior

Exponential Functions Recall General form 𝒇 𝒙 =𝒂 𝒃 𝒙 +𝒌 a = initial value that determines the shape a > 1 stretch; 0 < a < 1 shrink; -a = reflection b = growth if the value is > 1 b = decay if the value is between 0 and 1 k = horizontal asymptote & vertical shift

Guided Practice: Example 1, continued Complete the table of values to create a graph of the function. 𝑓 𝑥 = 2 𝑥 x f(x) –2 –1 1 2 3.4.2: Graphing Exponential Functions

Domain The collection of all x-values (inputs). For exponential functions the domain will always be all real numbers ℝ. Example: 𝒇 𝒙 = 𝟐 𝒙 Domain = all real numbers because any number can be used as x.

Range The collection of all y-values (outputs). +a: Range is all numbers > asymptote. -a: Range is all numbers < asymptote. Example: 𝒇 𝒙 = 𝟐 𝒙 Domain = all numbers > asymptote. y > 0

What happens at the ends of the graph. End Behavior What happens at the ends of the graph. Exponential functions have 2 end behaviors. One towards + or - infinity and one towards the horizontal asymptote. Example: 𝒇 𝒙 = 𝟐 𝒙 Left: As x → -∞, y → 0 Right: As x → +∞, y → +∞

Guided Practice: Example 2, continued Complete the table of values to create a graph of the function. 𝑓 𝑥 = 1 2 𝑥 x f(x) –2 –1 1 2 3.4.2: Graphing Exponential Functions

Example 2: 𝑓(𝑥) = 1 2 𝑥 Recall Not a reflection Decay Horizontal Asymptote: y = 0 y-intercept: (0, 1) Domain = _____________ Range = all numbers ____ asymptote y ____ _____ End behavior: Left: As x → -∞, y → ___ Right: As x → +∞, y → ___

Guided Practice: Example 3, continued Complete the table of values to create a graph of the function. 𝑓 𝑥 = 3 𝑥 +1 x f(x) –2 –1 1 2 3.4.2: Graphing Exponential Functions

Example 3: 𝑓 𝑥 = 3 𝑥 +1 Recall Not a reflection Growth Horizontal Asymptote: y = 1 y-intercept: (0, 2) Domain = _____________ Range = all numbers ____ asymptote y ____ _____ End behavior: Left: As x → -∞, y → ___ Right: As x → +∞, y → ___

Guided Practice: Example 4, continued Complete the table of values to create a graph of the function. 𝑓 𝑥 =−1( 2) 𝑥 +3 x f(x) –2 –1 1 2 3.4.2: Graphing Exponential Functions

Example 4: 𝑓 𝑥 =−1( 2) 𝑥 +3 Recall Reflection Decay Horizontal Asymptote: y = 3 y-intercept: (0, 2) Domain = _____________ Range = all numbers ____ asymptote y ____ _____ End behavior: Left: As x → -∞, y → ___ Right: As x → +∞, y → ___

Summarizing Strategy: Example for Absent friend Your absent friend needs you to show them an example of what they missed. Choose 3 of the following 5 features to identify for this exponential function: f(x) = 3x – 2 Asymptote y-intercept Domain Range End Behavior