Growth rate vs. Decay rate

Graphing Exponential Functions 𝑓 𝑥 =𝑎 𝑏 𝑥 If b > 1, the function shows growth If 0 < b < 1, the function shows decay The y-int. is the starting point or “a” The x-axis is the asymptote for these functions Choose several points, (-2, -1, 0, 1, 2), choose the appropriate scale and plot the points

Graphing Exponential Functions 𝑓 𝑥 =3 2.5 𝑥 Find the value of the base: 2.5 Does the function show growth or decay? Growth Make a table Graph the function Choose your scale carefully!

Graphing Exponential Functions 𝑓 𝑥 = .4 𝑥 Find the value of the base: 0.4 Does the function show growth or decay? Decay Make a table Graph the function Choose your scale carefully!

Practice Graphing Exponential Functions 𝑓 𝑥 = 1 2 𝑥 Find the value of the base: 1/2 Does the function show growth or decay? Decay Make a table Graph the function Choose your scale carefully!

Practice Graphing Exponential Functions 𝑓 𝑥 = 5 2 𝑥 Find the value of the base: 5/2 Does the function show growth or decay? Growth Make a table Graph the function Choose your scale carefully!

Practice Graphing Exponential Functions 𝑓 𝑥 =4 2 𝑥 Find the value of the base: 2 Does the function show growth or decay? Growth Make a table Graph the function Choose your scale carefully!

Exponential Decay Models 𝑓 𝑥 =𝑎 𝑏 𝑥 One of the most important factors is the bounciness or elasticity of a ball. For example, if a new golf ball is dropped onto a hard surface, it should rebound to about 2 3 of its drop height. Now suppose a new golf ball drops downward from a height of 27 feet onto a paved parking lot and keeps bouncing up and down, again and again. What is the decay factor or rate? 2 3 What is the y-intercept or starting point? 27 What is the function rule? 𝑦=27 2 3 𝑥

Exponential Decay Models 𝑓 𝑥 =𝑎 𝑏 𝑥 𝐷𝑒𝑐𝑎𝑦 𝑓𝑎𝑐𝑡𝑜𝑟=(1−𝑏) where b is the percent decrease You bought a car for $25,000, but its value is depreciating at a rate of 10% per year. What is the percent decrease (decay percent)? 0.10 What is the decay rate (factor)? 𝑏=(1−0.10)= .90 What is the y-intercept or starting point? 25000 What is the function rule? 𝑓 𝑥 =25000 .90 𝑥

From Table to Function Rule Linear

From Table to Function Rule Quadratic The differences are not constant. The y-values will repeat in a U-shape around the vertex. If it appears to be quadratic, try using the vertex form of the equation to test points.

From Table to Function Rule Exponential We can also look at the ratio: 𝟐𝒏𝒅 𝒚−𝒗𝒂𝒍𝒖𝒆 𝟏𝒔𝒕 𝒚−𝒗𝒂𝒍𝒖𝒆 if it is constant, the function is exponential.