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An exponential function is characterized by a constant base raised to a variable exponent. In simpler terms, it’s a type of function where the variable appears as the exponent. These functions are commonly written in the form:

• \( f(x) = a imes b^x \)

Here, 'a' is a constant that influences the starting value or vertical stretch of the graph, while 'b' is the base which determines the rate of growth or decay.

If \( b > 1 \), the function represents exponential growth, which means the values increase rapidly. However, if \( 0 < b < 1 \), like our example with \( b = \frac{1}{2} \), it portrays exponential decay, causing the values to decrease rapidly as \( x \) increases.

In the given function \( f(x) = 2\left(\frac{1}{2}\right)^{x} \), \( 2 \) is the initial quantity and \( \frac{1}{2} \) is the base highlighting a decay, suggesting a halving process every step further along the x-axis.

Understanding how to read and interpret the graph of a function is a key step in visual learning and problem-solving.

• The y-intercept is a crucial point on the graph. It is where the line or curve crosses the y-axis (vertical axis).

This point represents the output value of the function when \( x = 0 \). For instance, in our function \( f(x) = 2\left(\frac{1}{2}\right)^{x} \), substituting \( x = 0 \) gives us the y-intercept.

When interpreting the graph of an exponential function, it is important to observe how it behaves as \( x \) increases or decreases:

• Exponential growth graphs will rise rapidly.
• Exponential decay graphs, like this one, will fall as \( x \) grows.

Thus, knowing the y-intercept allows you to establish a starting point on the graph and understand how the function evolves from there.

The substitution method is a fundamental strategy used in mathematics, especially useful for finding the value of one variable in terms of another. It involves replacing a variable with a given value or expression. In the context of this exercise, we utilized substitution to find the y-intercept.

• Given a function \( f(x) = 2\left(\frac{1}{2}\right)^{x} \), to find its y-intercept:

We substitute \( x \) with 0, simplifying the problem to evaluate \( f(0) \). This method simplifies the expression because any quantity raised to the power of zero is 1.

So, performing the substitution:

• \( f(0) = 2\left(\frac{1}{2}\right)^{0} = 2 \times 1 \)

Concludes with the y-intercept being 2 at the point (0,2).

This method not only simplifies finding specific values but also helps verify solutions by substituting them back into equations.