Exam Questions - Exponential Type kepx (exponential types)

Exam Questions - Exponential Type kepx (exponential types)

Exponential Type ke^px Revision Content

Understanding Exponential Functions

  • An exponential function is one where the variable is in the exponent.
  • The type ke^px is specifically addressed here where k and p are constants.
  • These constants can be any real number.
  • The base, ‘e’, represents Euler’s number which is approximately 2.71828.
  • Exponential functions like these are crucial in modelling many real world phenomena such as population growth, radioactive decay and compound interest.

Evaluating Exponential Functions

  • To evaluate an expression of the type ke^px for a given value of x, substitute the value of x into the expression.
  • Use the rules of indices to simplify the expression where necessary.
  • A common rule often used is that e^0 equals 1, regardless of the values of k and p.

Graphing Exponential Functions

  • The functions of the type ke^px are always increasing or decreasing, they never stay constant.
  • The function will be increasing if p is positive and decreasing if p is negative.
  • The larger the absolute value of p, the faster the function increases or decreases.
  • The graph of the function never touches the x-axis, but gets infinitely closer as x approaches negative infinity - this is known as an asymptote.

Transformations of Exponential Functions

  • The constant k multiplies the output of the function and therefore results in a vertical stretch or shrink of the graph.
  • If k is negative, the function undergoes a vertical reflection, meaning it will be reflected in the x-axis.
  • Similarly, the constant p, if negative, can be thought of as causing a horizontal reflection of the function.

Solving Exponential Equations

  • Equations of the type ke^px = c (where c is some constant value) can be solved by first isolating the exponential part.
  • Next, apply the natural logarithm (ln) to both sides of the equation.
  • This allows simplification using the logarithm property ln(e^x) = x.
  • Solve the resulting equation for x.