Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-6 :\) a. Sketch the graph of each function. b. On the same set of axes, sketch the graph of the image of the reflection in the \(y\) -axis of the graph drawn in part a. . Write an equation of the graph of the function drawn in part b. $$ y=\left(\frac{3}{4}\right)^{x} $$

Step-by-Step Solution

Answer

Reflect the graph of \( y = \left( \frac{3}{4} \right)^x \) across the \( y \)-axis to get \( y = \left( \frac{3}{4} \right)^{-x} \).

Step 1: Understand the Function

The given function is an exponential function, specifically \( y = \left( \frac{3}{4} \right)^x \). This function represents an exponential decay since \( \frac{3}{4} \) is less than 1. As \( x \) increases, \( y \) decreases towards zero.

Step 2: Sketch the Original Graph

Plot the graph of \( y = \left( \frac{3}{4} \right)^x \). For some values: when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y = \frac{3}{4} \); when \( x = 2 \), \( y = \left( \frac{3}{4} \right)^2 = \frac{9}{16} \); and as \( x \) moves to negative values, \( y \) increases beyond 1, because negative exponents represent reciprocals.

Step 3: Reflect the Function in the y-axis

To reflect the graph in the \( y \)-axis, replace \( x \) with \( -x \) in the equation. The reflection of \( y = \left( \frac{3}{4} \right)^{x} \) in the \( y \)-axis is \( y = \left( \frac{3}{4} \right)^{-x} \). The graph will now show exponential growth, increasing as \( x \) does.

Step 4: Sketch the Reflection Graph

Draw the graph of \( y = \left( \frac{3}{4} \right)^{-x} \) on the same set of axes. This is the reflection of the original function, and it represents exponential growth instead of decay. For outputs, when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y = \left( \frac{3}{4} \right)^{-1} = \frac{4}{3} \); when \( x = 2 \), \( y = \left( \frac{3}{4} \right)^{-2} = \frac{16}{9} \).

Key Concepts

Graph Reflection

When we talk about the graph reflection of a function, we mean mirroring the graph across an axis. In the original exercise, the function given is an exponential decay, represented by \( y = \left( \frac{3}{4} \right)^x \). To reflect this graph across the \( y \)-axis, we replace \( x \) with \( -x \) in the equation.

• The original equation is \( y = \left( \frac{3}{4} \right)^x \).
• The reflected equation becomes \( y = \left( \frac{3}{4} \right)^{-x} \), which indicates a change from exponential decay to exponential growth.

Exponential functions reflect symmetrically about the \( y \)-axis when \( x \) is negated. This kind of transformation helps us understand different behavior types in functions—like switching from decreasing to increasing—by observing their graphical representations.

Exponential Growth

Exponential growth in functions occurs when a quantity increases rapidly over time. In the context of the given exercise, reflecting the exponential decay graph across the \( y \)-axis turns it into an exponential growth function.

• The original function \( y = \left( \frac{3}{4} \right)^x \) was a decay because the base \( \left( \frac{3}{4} \right) \) is less than 1.
• The reflected function \( y = \left( \frac{3}{4} \right)^{-x} \) now has an effective base of \( \left( \frac{4}{3} \right) \), which beats 1, thus showing growth.

In this growth scenario, as \( x \) increases, \( y \) also increases—meaning values of \( y \) become larger very quickly. Graphically, this translates to the curve rising steeply upwards as you move from left to right along the \( x \)-axis. Exponential growth is common in scenarios like population growth, where increases compound over time.

Exponential Decay

Exponential decay describes a process where quantities decrease rapidly, approaching zero. In the original scenario, the function \( y = \left( \frac{3}{4} \right)^x \) is an exemplary model of such behavior.

• Decay occurs here because each successive value of \( x \) sees \( y \) shrinking towards zero as the base \( \left( \frac{3}{4} \right) \) is less than 1.
• At \( x = 0 \), the function starts at an initial point of \( y = 1 \), illustrating that when no time has passed, the quantity is at its full initial measure.

This pattern can commonly be seen in natural decay processes, like how milk spoils or radioactive substances decrease over time. Although the exponential decay graph never quite touches the \( x \)-axis, it visually approaches it as \( x \) increases, exemplifying the rapid decrease tendency characteristic of exponential decay.