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Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. In mathematical terms, you have an exponential function like \(f(x) = a \cdot b^{x}\). Here, \(a\) is the initial value, while \(b\) is the base of the exponential function. If \(0 < b < 1\), the function exhibits exponential decay, meaning it shrinks as \(x\) increases.

In the example function \(f(x) = 3\left(\frac{1}{2}\right)^{x}\), the base \(\frac{1}{2}\) is less than one, which means the function's values decrease as \(x\) gets larger. This shows classic exponential decay behavior.

Real-world examples include radioactive decay and cooling of a hot object, where you notice the value dwindling over time.

Graphing functions help us visualize mathematical relationships and understand how they behave on a plane. With exponential functions, like \(f(x) = 3\left(\frac{1}{2}\right)^{x}\), you can graph the function by calculating and plotting key points.

Some important points include the y-intercept and additional values like \(f(1) = 1.5\) and \(f(2) = 0.75\). Plotting these calculated points on a graph allows you to draw the curve representing the function's trend. This curve will typically start at a high value and decrease asymptotically towards the x-axis without ever crossing it.

Tools like graph paper or graphing calculators make plotting easier and provide an accurate representation of the function's behavior over a range of values.

In mathematics, reflecting a function involves changing its orientation across an axis. For reflections across the \(y\)-axis, replace \(x\) with \(-x\) in the function's formula. This operation generates a new function, which is the mirror image of the original function on the graph.

Consider the function \(f(x) = 3\left(\frac{1}{2}\right)^{x}\). To reflect it across the \(y\)-axis, the transformation becomes \(f(-x) = 3(2)^{x}\), creating a mirror image of the original.

By calculating values such as \(f(-0) = 3\), \(f(-1) = 6\), and \(f(-2) = 12\), you can add these points to the graph where the reflection appears. This visualization helps students better grasp how functions modify and transform visually.

The \(y\)-intercept is a fundamental concept in graphing, where a function intersects the \(y\)-axis. It is the function's value when \(x = 0\). This point gives a starting reference for the graph and simplifies understanding how the function behaves.

For the exponential function \(f(x) = 3\left(\frac{1}{2}\right)^{x}\), set \(x = 0\) to find the \(y\)-intercept. The result is \(3 \times \left(\frac{1}{2}\right)^{0} = 3\), which means the function intersects the \(y\)-axis at \(y = 3\).

Similarly, the reflected function \(f(-x)\) also has its \(y\)-intercept at \(3\). Knowing the \(y\)-intercept helps identify important details of a graph and provides an easy point for initial plotting.