Exponential Graphs

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What is an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a * b^x, where a is a constant, b is a positive real number, and x is the variable. The base b determines the growth or decay rate.

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What does the graph of an exponential function look like?

Back

The graph of an exponential function is a smooth curve that either increases or decreases rapidly. It approaches the x-axis but never touches it (asymptote) and passes through the point (0, a) where a is the y-intercept.

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What is the significance of the base in an exponential function?

Back

The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

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How do you identify vertical shifts in exponential functions?

Back

Vertical shifts in exponential functions are indicated by adding or subtracting a constant from the function. For example, f(x) = b^x + k shifts the graph up by k units if k > 0 and down by k units if k < 0.

Front

What does the transformation 'Left 5' mean in the context of exponential functions?

Back

A transformation 'Left 5' means that the graph of the function is shifted 5 units to the left on the x-axis. This is represented by replacing x with (x + 5) in the function.

Front

What is the y-intercept of an exponential function?

Back

The y-intercept of an exponential function f(x) = a * b^x is the value of f(0), which equals a. This is the point where the graph crosses the y-axis.

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How do you determine the horizontal asymptote of an exponential function?

Back

The horizontal asymptote of an exponential function is determined by the constant added or subtracted from the function. For f(x) = b^x + k, the horizontal asymptote is y = k.

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What is the effect of changing the base of an exponential function?

Back

Changing the base of an exponential function affects the steepness of the graph. A larger base results in a steeper curve, while a smaller base (greater than 1) results in a flatter curve.

Front

How do you solve for h in the equation y = 2^(x-h)?

Back

To solve for h, compare the graph of y = 2^x with y = 2^(x-h). The value of h represents a horizontal shift. If the graph shifts to the right, h is positive; if it shifts to the left, h is negative.

Front

What is the general form of an exponential decay function?

Back

The general form of an exponential decay function is f(x) = a * b^x, where 0 < b < 1. This indicates that the function decreases as x increases.