In Exercises 61–64, give the equation of each exponential function whose graph is shown.

Verified step by step guidance

1. Identify the general form of an exponential function: y = ab^x.
2. Use the given points to set up equations. For the first graph, use points (0, 2), (1, 4), and (2, 8). For the second graph, use points (0, 1), (1, 4), and (2, 16).
3. For the first graph, substitute the point (0, 2) into the equation y = ab^x to find a. For the second graph, substitute the point (0, 1) into the equation y = ab^x to find a.
4. For the first graph, substitute the points (1, 4) and (2, 8) into the equation y = ab^x to create a system of equations and solve for b. For the second graph, substitute the points (1, 4) and (2, 16) into the equation y = ab^x to create a system of equations and solve for b.
5. Combine the values of a and b to write the final exponential function for each graph.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay, depending on the value of 'b'. Understanding their general form is crucial for identifying specific exponential functions from their graphs.

Graphing Points

To derive the equation of an exponential function from its graph, it is essential to analyze key points on the graph. For instance, the points (0,2), (1,4), and (2,8) indicate specific values of the function at those x-coordinates. These points can be used to set up equations that help determine the constants 'a' and 'b' in the exponential function.

Finding the Base and Initial Value

In an exponential function, the initial value 'a' is the output when x = 0, while the base 'b' can be found by examining the ratio of outputs at consecutive x-values. For example, if f(0) = 2 and f(1) = 4, the base can be calculated as b = f(1)/f(0) = 4/2 = 2. This process is vital for constructing the correct equation of the exponential function.