EXPONENTIAL FUNCTIONS Look at the two functions: –f(x) = x 2 g(x) = 2 x Each has a number raised to a power but reversed. Let’s evaluate each one for x = 0, 1, 2, 3, 4, 5, 10, 15 Are these increasing or decreasing? Which one changed the fastest? The function with x as the exponent changed much faster – increased exponentially. f(x) = ab x b: baseb ≠ 1

EXPONENTIAL: f(x) = ab x What happens to the exponential function if x = 0? Plug into your calculator: 2 0, 4 0, -3 0 Each time, the answer was 1 Anything raised to zero = 1 so, f(0) = a For exponential functions, x is often time a is often referred to as the “initial value” because when x = 0, the function = a. The initial value is what value you have before any thing starts happening.

NEGATIVE EXPONENTS Look at the two functions again, –f(x) = x 2 g(x) = 2 x Now, evaluate them for x = 0, -1, -2, -3, -4 Are they still increasing? When the exponent is negative it means the function is now in the denominator of a fraction. As x goes more negative, the denominator increases exponentially but the function gets small at a very fast pace In fact, it will start to approach 0. It will not look like a polynomial

EXPONENENTS - FRACTIONS Fractions as exponents have another meaning too. Plug into your calculator: 4 (1/2), 16 (1/2), 36 (1/2) The answer was always the square root. Try: 8 (1/3), 27 (1/3), 64 (1/3) This time the answer was always the cubed root. Fractions indicate a root. Luckily, your calculator will allow you to put in anything as the exponent. What does 2 6/3 mean?

FIND f(x) = 2 x f(3/2) = f(-3/2) = f(8) = f(-8) =

WHAT HAPPENS IF THE BASE CHANGES? f(x) = 2 x g(x) = (1/2) x Look at the two graphs. Are they increasing or decreasing? Why? The first one starts close to zero at grows to infinity. The second one starts at infinity and decays to close to zero. Growth function: Base > 1 Decay function: 0 < Base < 1 Base can’t be 1 Negative Base just means it is reflected across the x- axis

Base e There is a special base called, e. It is on your calculator It is like pi that it has a specific value e = The basic exponential function is: –f(x) = e x This models lots of phenomenon

EXPONENTIAL TRANSFORMATIONS We have already talked about negative bases reflecting across the x-axis. That is because we multiplied the whole function by -1 and did a vertical reflection. All the transformation rules will apply with the exponential function. When we had a negative exponent (multiplied just x by -1) it reflected horizontally. a is the initial value – it is the vertical dilation piece. g(x) = e 2x h(x) = e -x k(x) = 3e x

LOGISTIC FUNCTIONS Sometimes, it is not logical to think of a function increasing forever exponentially. There is a limit to how far the function can increase. Logistic functions are “capped” exponential functions. They limit the growth c is the limit

Money: You invested $2000 in a bank account earning 3% interest per year. Write the exponential function for this. How much money do you have after 3 years? After 15 years?

MATH AND FINANCE Money is compounded at an interest rate – the interest is normally assumed to compound annually – once per year The interest rate must match the period that is compounded. Compounded annually – annual interest rate Compounded monthly – annual interest rate/ 12

MONEY PROBLEMS Do the previous problem again, but assume that the interest is compounded monthly. What changes? Now how much do you have after 3 yrs, 15 yrs?

You have a culture of 100 bacteria that doubles every hour. Write the exponential function for this. How many bacteria do you have after 4 hours? When do you have 350,000 bacteria?

POINT-RATIO FORM Just like with linear functions, quadratic functions, polynomial functions, exponential functions can be written in a different form. The initial value, a, is the value of y when x = 0. This is the y-intercept Point-Ratio Form:

POINT-RATIO FORM Just like with linear functions, quadratic functions, polynomial functions, exponential functions can be written in a different form. The initial value, a, is the value of y when x = 0. This is the y-intercept Point-Ratio Form:

POINT-RATIO FORM Just like with linear functions, quadratic functions, polynomial functions, exponential functions can be written in a different form. The initial value, a, is the value of y when x = 0. This is the y-intercept Point-Ratio Form:

POINT-RATIO FORM Just like with linear functions, quadratic functions, polynomial functions, exponential functions can be written in a different form. The initial value, a, is the value of y when x = 0. This is the y-intercept Point-Ratio Form:

USING THE SOLVER Your calculator has the ability to solve some equations for you. Go to MATH Go to Solver Scroll Up to get the equation screen Put in your equation = 0 Hit enter Put in a guess for the answer (usually 0) Hit Alpha Enter (Solve)