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Reflecting a function on the y-axis essentially means flipping it horizontally. This transforms each point of the function to its mirror image concerning the y-axis. Let's explore this with the exponential function provided: - The original function is \( f(x) = 3 \left( \frac{1}{2} \right)^x \).- To reflect it across the y-axis, we replace \( x \) by \(-x\) in the function, giving us \( f(-x) = 3 \times 2^x \).Here's how reflection affects key characteristics:- **The curve's direction**: A reflection changes a decreasing function into an increasing one, and vice versa.- **The shape of the graph**: Remains the same distance from the y-axis on either side.Reflection is particularly useful for visualizing symmetrical behavior or for comparing function behavior under transformations.

Graphing functions can seem daunting, but by following a systematic approach, it becomes a straightforward task. Here's a simple breakdown:1. **Selecting Points to Plot**: - Choose x-values that are easy to work with, such as 0, 1, -1, 2, etc. - For each x-value, calculate the corresponding y-value using the function. - For example, in our original function, \( f(x) = 3 \left( \frac{1}{2} \right)^x \), plug in values such as \( x = 0, 1, -1\) to get the y-values.2. **Plotting the Points**: - Mark these points on a coordinate plane (a grid where you plot graphs). - For a function like \( f(x) = 3 \left( \frac{1}{2} \right)^x \), you would plot points like (0, 3), (1, 1.5), etc.3. **Sketching the Curve**: - Once you've marked enough points, draw a smooth curve through them. - This will show the trend of the function, either increasing, decreasing, or following another pattern.Utilizing these techniques, even complex functions can be broken down into manageable parts, providing a clear visual understanding of function behavior.

Finding the y-intercept of a function involves determining where the function crosses the y-axis. By definition, the y-axis is where \( x = 0 \).- **For an exponential function** like \( f(x) = 3 \left( \frac{1}{2} \right)^x \), substitute \( x = 0 \) into the function: - Calculating this gives \( f(0) = 3 \times 1 = 3 \).- **For the reflected function** \( f(-x) = 3 \times 2^x \), do the same substitution: - Calculating \( f(-0) = 3 \times 1 = 3 \) confirms the y-intercept is also 3.This simple substitution provides quick insight, making the y-intercept one of the easiest points to determine. When graphing, the y-intercept is critical for accurately placing the function on the coordinate plane.