MCR3U1 U6L4 EXPLORING THE PROPERTIES OF EXPONENTIAL FUNCTIONS (An Investigation) PART A ~ DEFINITION An exponential function is a function of the form: 𝑦 = 𝑎(𝑏)𝑥 PART B ~ GRAPHING EXPONENTIAL FUNCTIONS Graph the exponential functions, 𝑦 = 2𝑥 and 𝑦 = 2−𝑥 , on the given grid. y x 𝑦=2 𝑥 –3 –2 –1 0 1 2 3 x 1 𝑥 𝑦=( ) 2 –3 –2 –1 0 1 2 3 x PART C ~ PROPERTIES OF EXPONENTIAL FUNCTIONS PROPERTIES 𝑦 = 2𝑥 1 𝑥 𝑦 = 2−𝑥 or 𝑦 = (2) domain range y–intercept equation of the asymptote increasing/decreasing function The exponential function, 𝑦 = 𝑏 𝑥 , is: an increasing function representing growth when b > 1 an decreasing function representing decay when 0 < b < 1 MCR3U1 U6L4 PART D ~ COMPARING FIRST DIFFERENCES OF VARIOUS FUNCTIONS Linear, quadratic, and exponential functions have unique first–difference patterns that allow them to be recognized. Calculate the first–differences for each of the following functions: Function: LINEAR x 1 2 3 4 5 QUADRATIC 𝑦 = 2𝑥 x 1 2 3 4 5 EXPONENTIAL 𝑦 = 𝑥2 𝑦 = 2𝑥 x 1 2 3 4 5 Patterns Observed in the First–differences: Sketch each of the following functions on the same set of axes: y 𝑦 = 3𝑥 , 𝑦 = 4𝑥 , 𝑦 = 5𝑥 𝑦 = 1𝑥 1 𝑥 1 𝑥 1 𝑥 𝑦=( ) , 𝑦=( ) , 𝑦=( ) 3 4 5 Why do all exponential graphs share a POI of (0, 1)? 1 x HOMEWORK: p.243 #1, 2