Algebra 1 - Exponential Functions


  • The exponential function is a function in the form of f(x)=bx or y=bx where b>0 and b1.
  • The domain of f is the set of all real numbers, and the range is the set of all positive real numbers.
  • The line y=0 (x-axis) is the horizontal asymptote of f(x)=bx.
  • If the function is increasing, it is in the form f(x)=bx where b>0 and b1.
  • If the function is decreasing, it is in the form of f(x)=bx where 0<b<1.
  • The graph of f(x)=bx where b>0 and b1 is shown below:

  • The graph of f(x)=bx where 0<b<1 is shown below:

Natural Exponential Function, Exponential Growth, and Exponential Decay

  • Natural Exponential Function is a function in the form of fx=ex where e=Euler's number.
  • Euler's number is an irrational number that is approximately equal to 2.71828 or 2.72.
  • This function has the same form of graph and properties as those with y=bx where b>0 and b1.
  • The inverse of the natural exponential function is the natural logarithmic function. 


Exponential Growth

  • A mathematical model of population increase is given by y=y0ert , where y is the population after t years, y0 is the present population, and r is the population growth rate. 
  • For the doubling time growth model, the equation is y=y02tT, where y0 is the present population, y is the population after t years, T is the doubling, and t is the time in years.
  • Another form of exponential growth is the concept of compound interest given by the formula y=y01+rkkt where y0 is the initial amount, y is the value at the end of t years, r is the nominal rate, and k is the number of conversions.
  • The appreciation formula is given by y=y01+rt where y is the increased value after t years, y0 is the initial value, and r is the rate of increase.
  • The expression 1+r or 1+rk is called the growth factor.


Exponential Decay

  • This is modeled by the equation y=y0e-rt where r is the rate of decrease, e is Euler's number.
  • The half-life decay model is given by y=y012tT where T is the half-life.
  • The formula for depreciation is given by y=y01-rt where r is the annual rate of depreciation. 
  • The expression 1-r is called the decay factor.

Transformation of the Graph of Exponential Function

  • Transformation of the graph of an exponential function describes its relationship with the parent function y=bx
  • Reflection: The graph of y=-f(x) is the reflection about the x-axis of the graph of y=f(x). The graph of y=f(-x) is the reflection of the y-axis of the graph of y=f(x).
  • Stretching and Shrinking: In function f(x)=abx, if a<1 or a>1, the effect can be observed in the graph of the parent function f(x)=bx. If a>1, the graph shrinks. If 0<a<1, then the graph of f(x)=abx is wider. 


Vertical and Horizontal Shift: 

  • For function f(x)=bx-c, the graph of the parent function f(x)=bx shifts c units to the right since c>0.
  • For function f(x)=bx+c or f(x)=bx-(-c), the graph of the parent function f(x)=bx shifts c units to the left since c<0.
  • For function f(x)=bx-d, the graph of the parent function shifts d units downward since d<0. The horizontal asymptote is the line y=-d.
  • For function f(x)=bx+d, the graph of the parent function shifts d units upward since d>0. The horizontal asymptote is the line y=d.

Exponential Equations and Some Solved Examples

  • An exponential equation is an equation involving exponential expressions. The following are some examples. Notice that the exponent has variables. 
  • 2x=8
  • 52x-3=625
  • 17x=3432x-1
  • Use the One-to-One Property of Exponential Functions in solving exponential equations. Let b be a positive real number, with b1 and let x and y be real numbers. If bx=by, then x=y.


Example 1. Solve for x in the equation 2x=64.



Thus, we get x=6.


Example 2. Solve the equation 12x+7=64-35-x




Since there are common bases, we have:






Solved Examples of Exponential Growth and Exponential Decay

Example 1. The population of a particular city in Florida increases according to the exponential model y=4500e0.015t where t is in the years. What will the population be after seven years?



y4,998.20 or y5,000

Hence, it is predicted to have a total population of 5,000 after seven years.


Example 2. A toxic radioactive substance, Plutonium-230, has a half-life of 24,100 years. Suppose 5 mg of Plutonium was released in a nuclear accident; how much of the 5 mg will remain after 100 years?


Use the exponential decay model, y=y012tT


y4.985 mg

Hence, about 4.985 mg will be left after 100 years.

Cheat Sheet

  • There are two major forms of exponential functions, fx=bx where b>0, b1 (increasing) and fx=bx where 0<b<1 (decreasing).
  • The graphs of the exponential functions fx=2x and gx=2-x show a reflection in the y-axis.
  • The graphs of the exponential functions fx=2x and gx=-2x show a reflection in the x-axis.
  • The horizontal asymptote of the exponential function in the form of fx=bx is the line y=0 (x-axis).
  • The horizontal asymptote of y=2x-3+1 is the line y=1.
  • Using the one-to-one property of exponential functions implies that both expressions must have the same base to solve the equation.

Blunder Areas

  • The base b of an exponential function fx=bx is always positive. Don't be confused with the form fx=-bx.
  • An exponential function in the forms fx=bx or gx=b-x have no x-intercepts. However, the case of fx=bx-d has an x-intercept.
  • Always use the One-to-One Property of Exponential functions in solving exponential equations.
  • In the case of exponential equations wherein we cannot change both sides to similar bases, logarithms are applicable. Some examples of these equations are 72x-3=1843+4x=92x+1, or 15=28-3x.