Exponential function
Analyze exponential functions y=ab^x in Grade 10 algebra, identify growth (b>1) vs decay (0<b<1), find y-intercept, horizontal asymptote, and solve exponential equations.
Key Concepts
A function of the form $y = ab^x$ is an exponential function if $x$ is a real number, $a \neq 0$, $b 0$, and $b \neq 1$. The parent function of all exponential functions with base $b$ is $y = b^x$.
The function $y = 5^x$ shows exponential growth because its base $b=5$ is greater than 1. The function $y = (\frac{1}{3})^x$ shows exponential decay because its base $b=\frac{1}{3}$ is between 0 and 1. The expression $y = ( 4)^x$ is not an exponential function because its base is negative.
Think of 'b' as a growth multiplier. The variable 'x' in the exponent is the secret sauce that makes the function value skyrocket or shrink incredibly fast. This is why we call it exponential growth! The base 'b' must be a positive number; otherwise, you'd get a wacky, disconnected graph that isn't a true function.
Common Questions
What is the general form of an exponential function and what do the parameters mean?
f(x) = a·bˣ where a is the initial value (y-intercept when no shift), b is the base (growth factor). If b>1 it is exponential growth; if 0<b<1 it is exponential decay.
How do you identify exponential growth vs. decay from an equation?
If b>1 (e.g., y=3(2)ˣ), the function grows exponentially. If 0<b<1 (e.g., y=5(0.5)ˣ), it decays. The y-values either increase or decrease without bound.
What is the horizontal asymptote of y=2·3ˣ?
y=0 (the x-axis). As x→-∞, 3ˣ→0, so y→0. The function never crosses y=0 but approaches it from above.