The presence of this doubling time or half-life is characteristic of exponential functions, indicating how fast they grow or decay. The function machine metaphor is useful for introducing parameters into a function. We could capture both functions using a single function machine but dials to represent parameters influencing how the machine works.
We could think of a function with a parameter as representing a whole family of functions, with one function for each value of the parameter. We can also change the exponential function by including a constant in the exponent. Therefore, every y value in the range corresponds to only one x value.
So, for any particular value of y, you can use the graph to see which value of x is the input to produce that y value as output. Because for each value of the output y, you can uniquely determine the value of the corresponding input x, thus every exponential function has an inverse function. The inverse function of an exponential function is a logarithmic function, which we will investigate in the next section.
Compare the graphs of these functions. Defining an Exponential Function A study found that the percent of the population who are vegans in the United States doubled from to Percent change refers to a change based on a percent of the original amount. Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time. Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain. Figure 1. Identifying Exponential Functions Which of the following equations are not exponential functions? Show Solution By definition, an exponential function has a constant as a base and an independent variable as an exponent. Try It Which of the following equations represent exponential functions? Show Solution Follow the order of operations.
Show Solution [latex]5. Exponential Growth A function that models exponential growth grows by a rate proportional to the amount present. Finding Equations of Exponential Functions In the previous examples, we were given an exponential function, which we then evaluated for a given input. How To Given two data points, write an exponential model. How To Given the graph of an exponential function, write its equation. First, identify two points on the graph.
Choose the y -intercept as one of the two points whenever possible. Try to choose points that are as far apart as possible to reduce round-off error.
Figure 5. Try It Find an equation for the exponential function graphed in Figure. Figure 6. How To Given two points on the curve of an exponential function, use a graphing calculator to find the equation. Press [STAT]. Clear any existing entries in columns L1 or L2. In L1 , enter the x -coordinates given. In L2 , enter the corresponding y -coordinates.
Press [STAT] again. Show Solution Follow the guidelines above. Try It Use a graphing calculator to find the exponential equation that includes the points 3, Applying the Compound-Interest Formula Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. Try It Refer to Figure.
Evaluating Functions with Base e As we saw earlier, the amount earned on an account increases as the compounding frequency increases. Investigating Continuous Growth So far we have worked with rational bases for exponential functions. Calculating Continuous Decay Radon decays at a continuous rate of Show Solution 3. Exponential Growth Function Compound Interest. A function is evaluated by solving at a specific value.
See Figure and Figure. An exponential model can be found when the growth rate and initial value are known. An exponential model can be found when the two data points from the model are known.
An exponential model can be found using two data points from the graph of the model. An exponential model can be found using two data points from the graph and a calculator.
The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known.
Continuous growth and decay models can be found when the initial value and growth or decay rate are known. Section Exercises Verbal Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function. Show Solution Linear functions have a constant rate of change. Show Solution When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account.
Algebraic For the following exercises, identify whether the statement represents an exponential function. The average annual population increase of a pack of wolves is Show Solution exponential; the population decreases by a proportional rate. Show Solution not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function.
Which forest had a greater number of trees initially? Algebra 1 Linear inequalities Overview Solving linear inequalities Solving compound inequalities Solving absolute value equations and inequalities Linear inequalities in two variables.
Algebra 1 Systems of linear equations and inequalities Overview Graphing linear systems The substitution method for solving linear systems The elimination method for solving linear systems Systems of linear inequalities. Algebra 1 Factoring and polynomials Overview Monomials and polynomials Special products of polynomials Polynomial equations in factored form.
Algebra 1 Quadratic equations Overview Use graphing to solve quadratic equations Completing the square The quadratic formula.
Algebra 1 Radical expressions Overview The graph of a radical function Simplify radical expressions Radical equations The Pythagorean Theorem The distance and midpoint formulas. Algebra 1 Rational expressions Overview Simplify rational expression Multiply rational expressions Division of polynomials Add and subtract rational expressions Solving rational expressions.
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