Question

In Exercises 11–16, describe the transformation of f represented by g. Then graph each function. (See Example 2.)\(f(x)=x^4, g(x)=(2 x)^4-3\)

Short answer

The transformation of f represented by g is a horizontal compression by a factor of 2 and a vertical shift downwards by 3 units.

Step-by-step solution

Identify the Basic Function

Recognize the basic function here. The basic function is \(f(x) = x^4\).

Evaluate g(x)

Evaluate the given function \(g(x) = (2x)^4 - 3\). Note that the term \(2x\) is raised to the power 4, which is like \(f(2x)\) and there is a minus 3 outside the brackets which shifts the plot 3 units downwards.

Describe the Transformations

The transformation of \(f\) represented by \(g\) can be described as a horizontal compression by a factor of 2 and a vertical shift downwards by 3 units.

Graph the Function

To graph the function \(f(x) = x^4\) and \(g(x) = (2x)^4 - 3\), plot some points for both functions and connect the dots smoothly. For \(f(x) = x^4\), the function increases rapidly for both the positive and negative x values. For \(g(x) = (2x)^4 - 3\), the function also increases rapidly on both sides but it is compressed horizontally to half its original width and shifted downwards by 3 compared to the graph of \(f(x) = x^4\).

Key concepts

Horizontal Compression

The concept of horizontal compression involves modifying a graph by compressing it along the x-axis. In the given exercise, the transformation from the basic function \(f(x) = x^4\) to \(g(x) = (2x)^4\) exemplifies this. To understand horizontal compression, note that a factor inside the function, such as the 2 in \((2x)\), affects the x-coordinates:

• Every x-value is multiplied by the reciprocal of this factor.
• A factor greater than 1, such as 2, causes the graph to compress, making it appear 'squished' horizontally.

This means that whatever distance an x-value is originally from the y-axis for \(f(x)\), it is half that distance for \(g(x)\). Consider replacing \(x\) with \(2x\), which makes the input double as fast, significantly narrowing the graph horizontally.

Vertical Shift

A vertical shift changes the position of a graph up or down along the y-axis. For the function \(g(x) = (2x)^4 - 3\), the "-3" represents a vertical shift. Here's how it works:

• Each point on the graph of the base function \(f(x) = x^4\) is simply moved three units down.
• The y-values are adjusted by subtracting 3 from each.

Thus, if a point \((x, y)\) is on \(f(x)\), the same point becomes \((x, y-3)\) on \(g(x)\). This simple transformation shifts the entire graph downward but retains the original shape of the curve.

Polynomial Function

Polynomial functions are mathematical expressions involving powers of \(x\). They are characterized by their smooth and continuous curves. In our exercise, the basic function \(f(x) = x^4\) is a polynomial.Key characteristics of polynomial functions include:

• The degree of the polynomial determines the maximum number of turns the graph can have. Here, \(x^4\) is a degree 4 polynomial.
• Polynomials are symmetric about the y-axis if all powers involved are even, as shown by \(x^4\).
• As x-values increase (positive or negative), the function value increases significantly because of the high degree.

Explore how different transformations like those in this exercise impact their graphs, crucial for mastering more complex mathematical concepts.

Graphing Functions

Graphing functions involves plotting points from a given equation onto a coordinate system and connecting them to visualize the function's shape.For graphing \(f(x) = x^4\) and \(g(x) = (2x)^4 - 3\):

• Start by selecting some x-values to calculate corresponding y-values for \(f(x)\) and plot these points.
• For \(g(x)\), remember the transformations: compress horizontally by 2 and shift vertically down by 3.
  • As you compute, say for x = 1, evaluate \(g(x) = (2 \times 1)^4 - 3\) to plot points on \(g(x)\).
• Smoothly connect your points to reveal the curves of both functions, noting differences due to transformations.
  • Graphing allows you to observe the change in shape, width, and position, brought by transformations.

Through graphing, one can see how compression and shift manifest visually, elucidating the transformation process.