Problem 12 - Contemporary Precalculus

Question

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$g(x)=-\left(2^{x}\right)$$

Step-by-Step Solution

Answer

Answer: To transform the graph of h(x) = 2^x into the graph of g(x) = - (2^x), we need to flip the graph of h(x) about the x-axis.

Step 1: Identify the base function and its transformation

The base function is h(x) = 2^x. The function we want to transform it into is g(x) = - (2^x). We can see that the transformation involves negating the output of the base function h(x).

Step 2: Describe the transformation

Since the function g(x) is the negation of h(x), the transformation required to get the graph of g(x) from the graph of h(x) is flipping the graph about the x-axis. This means that all the points on the graph of h(x) will have their y-coordinates negated, resulting in the flipped graph.

Step 3: List the transformations

To better visualize this, we can list all the individual transformations needed. In this case, there is only one transformation: 1. Flip the graph of h(x) about the x-axis. These steps show how to transform the graph of h(x) = 2^x into the graph of g(x) = - (2^x) by listing the transformations required to achieve this.

Key Concepts

flipping graphs

Flipping graphs is a type of transformation where the graph of a function is inverted to form a mirror image across a specified line, such as the x-axis or y-axis. In mathematics, this process is referred to as a reflection. When you flip a graph, the points on the graph change position based on the line of reflection. A flipped graph can look quite different from the original, though the shape remains the same. This transformation is particularly useful for visualizing how a graph behaves when reflected. Some key things to remember about flipping graphs:

• The basic shape of the graph doesn't change; it only appears as a mirror image.
• Flipping can occur over either the x-axis or y-axis, depending on the transformation.
• This transformation affects the output values (y-values) of the function.

For example, flipping the exponential graph of a positive function over the x-axis will make its y-values negative, thus inverting the graph vertically.

exponential functions

Exponential functions are a class of mathematical functions that involve the raising of a constant, known as the base, to a variable exponent. The typical form of an exponential function is given by \(f(x) = a^x\), where \(a>0\) and \(a eq 1\). These functions are characterized by their rapid rate of growth or decay, depending on whether the base \(a\) is greater or less than one. Exponential functions have several important properties:

• They are continuous and smooth, without any jumps or breaks.
• The graph of an exponential growth function \(f(x)=a^x\) with \(a>1\) rises upwards from left to right.
• Conversely, if \(0
• The y-intercept of \(f(x) = a^x\) is always at point (0,1), since \(a^0 = 1\).

Exponential functions frequently occur in real-world scenarios, such as population growth, radioactive decay, and interest calculations, making them crucial for various applications.

reflection over x-axis

Reflection over the x-axis is a specific graph transformation where each point of the graph is mirrored across the x-axis. This results in the inversion of the graph, transforming each y-value into its negative counterpart, while x-values remain unchanged.For instance, if you start with a function \(h(x) = 2^x\), reflecting it over the x-axis would yield \(g(x) = -2^x\). This transformation changes:

• Positive y-values to negative y-values.
• Negative y-values to positive y-values.
• The graph makes a vertical flip over the x-axis.

This reflection technique is useful for understanding symmetries in graphs, as well as manipulating the graph to fit certain criteria or model a specific scenario. While the x-values stay constant, this transformation fundamentally alters the visual presentation of the function without changing its domain. Therefore, reflecting over the x-axis is a powerful tool in graph transformation that helps visualize and explore the behavior of functions in new dimensions.