91Ó°ÊÓ

When you have a function, reflecting it about the y-axis is a simple yet important transformation. This reflection involves changing every occurrence of variable \( x \) to \( -x \). For our function \( g(x) = -2(0.25)^x \), the reflection would be \( h(x) = -2(0.25)^{-x} \). This can be simplified to \( h(x) = -2(4)^x \).
Reflections about the y-axis will essentially "mirror" the function horizontally. Think about your reflection in a virtual mirror placed on the y-axis; the points will appear on the opposite side at the same vertical level. This transformation does not affect the y-intercept of the function, as this point lies directly on the y-axis.

• Original function: \( g(x) = -2(0.25)^x \)
• Reflected function: \( h(x) = -2(4)^x \)

Finding the y-intercept of a function is straightforward. It is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). Therefore, to find the y-intercept, substitute \( x \) with \( 0 \) in the function.
For both our original function \( g(x) = -2(0.25)^x \) and its reflection \( h(x) = -2(4)^x \), we calculate:

• \( g(0) = -2(0.25)^0 = -2 \)
• \( h(0) = -2(4)^0 = -2 \)

This shows that both functions intersect the y-axis at \((0, -2)\). The y-intercept of a function can be visualized as the starting point of the graph on the y-axis when drawn.

Exponential functions are fascinating because of their rapid change over a short span of x-values. There are two behaviors to note: exponential growth and exponential decay.
In our original function \( g(x) = -2(0.25)^x \), the term \( (0.25)^x \) demonstrates exponential decay. This is because the base \( 0.25 \) is between 0 and 1, making the value shrink as x increases. This shrinking or decaying is further emphasized by the \(-2\), which flips the graph vertically over the x-axis.
Conversely, in the reflected function \( h(x) = -2(4)^x \), the base \( 4 \) is greater than 1, indicating exponential growth. Even though it grows, the \(-2\) in front flips the result, leading the graph to decrease in y-values as x increases.

• Exponential decay: \( 0 < \text{base} < 1 \)
• Exponential growth: \( \text{base} > 1 \)

Graph transformations can take various forms such as translations, reflections, stretches, and compressions. These transformations alter how a function's graph appears.
Starting with the basic function form \( a(b)^x \):

• Reflection: Changing \( x \) to \( -x \) reflects the graph about the y-axis.
• Scaling vertically affects the steepness, either stretching or compressing it.
• Translation moves the graph horizontally or vertically, but not applicable to our function \( g(x) = -2(0.25)^x \).

Our original function exhibits reflection and vertical scaling:

• Reflect through the negative sign in \(-2\)
• Y-axis reflection by substituting \( x \) to \( -x \) in reflection.

These transformations help in visualizing and understanding the behavior of functions in relation to their algebraic expressions.