Write an exponential function for the graph that passes through the given points. (0,-5) and (-

step 1 Okay, again, I want to write an equation for the exponential graph this time through 0 -9 -5 and

Write an exponential function for the graph that passes...

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Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form f(x) = a · b^x, where 'a' represents the initial value (the value of the function when x is zero) and 'b' is the base, which determines how the function grows or decays. These functions are used to model processes where the rate of change is proportional to the current value, such as population growth, radioactive decay, or compound interest.

Determining Parameters Using Given Points

When constructing an exponential function from points on its graph, each point provides an equation by substituting its coordinates into the general form. The point at x = 0 is particularly useful because it immediately gives the value of 'a', as f(0) = a · b^0 simplifies to f(0) = a. With 'a' known, another point can be used to solve for 'b' by substituting its x and y values into the function.

Solving for the Base

After identifying the initial value from the y-intercept, the next step involves isolating the base 'b' in the exponential equation. This is done by substituting another set of coordinates into the function and solving the resulting equation for 'b'. The base is crucial as it indicates whether the function shows exponential growth (b greater than 1) or decay (b between 0 and 1).

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