Problem 3 - 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. $$ y=4^{x} $$

Step-by-Step Solution

Answer

The reflected graph's equation is \(y=4^{-x}\), representing the reflection of \(y=4^x\) over the \(y\)-axis.

Step 1: Graph the Original Function

We start by sketching the graph of the function \(y=4^{x}\). This is an exponential function with base 4. At \(x=0\), \(y=4^0=1\). For \(x>0\), the function increases rapidly, while for \(x<0\), the function approaches zero but never actually reaches it. Key points on the graph include \((0,1), (1,4), (2,16), (-1,0.25), (-2,0.0625)\), and so forth.

Step 2: Identify the Reflection over the y-axis

To find the reflection of the function \(y=4^{x}\) across the \(y\)-axis, we need to replace \(x\) with \(-x\) in the equation of the function. This results in the function \(y=4^{-x}\).

Step 3: Graph the Reflected Function

Next, plot the graph of \(y=4^{-x}\). This graph is a reflection over the \(y\)-axis of the original function \(y=4^x\). Key points on this graph include \((0,1), (-1,4), (-2,16), (1,0.25), (2,0.0625)\). It decreases rapidly for \(x>0\) and approaches zero for \(x<0\).

Step 4: Write the Equation of the Reflected Graph

The equation representing the reflected graph is \(y=4^{-x}\). This represents the exponential decay version of its original by reflecting the original across the \(y\)-axis.

Key Concepts

Graphing Functions

Graphing exponential functions like \(y=4^x\) can be quite straightforward once you understand their basic characteristics. An exponential function displays rapid growth as you move along the x-axis. Let's break it down:

• The base of the function here is 4. This tells us how quickly the function will grow or shrink.
• At \(x=0\), any exponential function with a base greater than 1 gives a result of 1 since any number to the power of 0 is 1.
• For \(x>0\), the graph of \(y=4^x\) increases at an accelerating pace.
• Conversely, for \(x<0\), the values approach zero, but never become negative or reach zero, creating an asymptote at the x-axis.

To sketch \(y=4^x\), plot a few key points such as \((0,1), (1,4), (2,16), (-1,0.25), (-2,0.0625)\). See how the curve rises steeply in the positive x direction and gets flat on the negative side.

Reflections in the Coordinate Plane

Reflections in the coordinate plane are transformations that flip a graph across a specified line, like the x-axis or y-axis. In this exercise, we focus on reflecting the function \(y=4^x\) across the y-axis:

• To reflect a graph across the y-axis, substitute \(x\) with \(-x\) in the function's equation.
• Thus, the reflection of \(y=4^x\) becomes \(y=4^{-x}\).

Reflecting across the y-axis changes each point's x-coordinate: a point \((a, b)\) on the original graph transforms to \((-a, b)\) on the reflected graph. The behavior of the function also reverses:

• The reflected graph, \(y=4^{-x}\), will now decrease as \(x\) increases.
• Key points like \((0,1), (1,0.25), (2,0.0625)\) on \(y=4^{-x}\) complement those on \(y=4^x\) but in mirrored form.

Think of it like holding a mirror on the y-axis; the entire graph way appears flipped.

Exponential Growth and Decay

Exponential functions like \(y=4^x\) are excellent examples of how exponential growth and decay work. Here's how they manifest in this exercise:

• **Exponential Growth**: The function \(y=4^x\) epitomizes exponential growth. As \(x\) increases, the value of \(y\) grows rapidly. This is characteristic of growth models in real life, such as population growth, where increase is proportional to the current state.
• **Exponential Decay**: Conversely, the reflected function \(y=4^{-x}\) exhibits exponential decay. Here, the function's values decrease fast as \(x\) increases. Scenarios such as radioactive decay, where quantities decrease by a consistent factor, exemplify this behavior.

Remember:

• Exponential growth means you multiply by the base repeatedly as \(x\) increases, leading to larger outputs.
• Exponential decay involves dividing by the base, resulting in a decline to smaller outputs.

Understanding whether the function is growing or decaying helps identify real-world implications and the effects of transformations like reflections.