Problem 64 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(x)=2 \sqrt{x}-2 x+0.25 x^{2}\). Using the same axes, draw the graphs of \(y=f(x), y=f(1.5 x)\), and \(y=\) \(f(x-1)+0.5\), all on the domain \([1,5]\).

Step-by-Step Solution

Answer

Draw all three transformations of the function on the same set of axes within the domain [1,5].

Step 1: Understanding the Functions

The given function is \(f(x) = 2 \sqrt{x} - 2x + 0.25x^2\). We need to graph three functions: \(y = f(x)\), \(y = f(1.5x)\), and \(y = f(x-1) + 0.5\) on the domain \([1,5]\).

Step 2: Graphing the Original Function

Firstly, calculate points for \(y = f(x) = 2 \sqrt{x} - 2x + 0.25x^2\). Evaluate the function for several values of \(x\) within \([1,5]\), and plot these points to draw the graph of \(y = f(x)\).

Step 3: Graphing the Compressed Function

For \(y = f(1.5x)\), it involves compressing the graph horizontally. To graph this, calculate \(f(1.5x)\) for several values within the range \([1,5]\). Use \(y = 2 \sqrt{1.5x} - 2(1.5x) + 0.25(1.5x)^2\) to find the new points.

Step 4: Graphing the Translated and Adjusted Function

For \(y = f(x-1) + 0.5\), shift the graph of \(f(x)\) right by 1 unit and up by 0.5 units. Calculate \(f(x-1) = 2 \sqrt{x-1} - 2(x-1) + 0.25(x-1)^2\) for \(x\) values in \([1,5]\) and add 0.5 to \(f(x-1)\) to plot this graph.

Step 5: Combining and Visualizing All Graphs

On the same set of axes, plot the graphs of \(y=f(x)\), \(y=f(1.5x)\), and \(y=f(x-1) + 0.5\) for \(x\) values from 1 to 5. Ensure that each graph is clearly labeled, perhaps with different colors or line styles, to distinguish between them.

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in algebra and calculus that involves plotting points on a set of axes to represent a function visually. In this exercise, you're dealing with the function

• \( f(x) = 2 \sqrt{x} - 2x + 0.25x^2 \).

The goal is to create graphs for three versions of this function. Each graph will be plotted on the same axes but with different transformations applied.

• The original function's graph, \( y = f(x) \), is plotted using values of \( x \) within the given domain \([1,5]\).
• For \( y = f(1.5x) \), we compress the graph horizontally. This means each point is pulled closer to the y-axis as compared to the original graph, making it narrower.
• With \( y = f(x-1) + 0.5 \), you're translating the function. This involves shifting the graph to the right and up. The graph moves right by 1 unit and upward by 0.5 units on each point.

Graphing can help you visually interpret how functions behave under transformations, revealing how the shapes shift or stretch across the axes. Remember, all these graphs are done partially by adjusting the formula used to calculate each point plotted on the axes.

Domain and Range

When dealing with functions, understanding the domain and range is crucial.

• The **domain** refers to all possible input values (\( x \)-values) that a function can accept.
• The **range** is the set of possible outputs (\( y \)-values) the function can produce.

In our case, the domain provided is \([1,5]\). This means you will only consider \( x \) values within this interval when evaluating the functions.
When you graph \( f(x) \), \( f(1.5x) \), and \( f(x-1) + 0.5 \), making sure that the domain is adhered to is important because any calculations and resulting points need to fall within this range. Though transformations might shift or scale the graph, these movements do not affect the domain given to you.While the domain in this exercise is clearly specified, figuring out the range could require further calculation or visual observation based on the plotted graphs. The transformation of the function could potentially shift the range, particularly in the vertically translated version of the function. Monitoring these changes through graphing remains an effective means to understanding.

Algebraic Manipulation

Algebraic manipulation is key to transforming functions into new forms to match desired graphing criteria. One of the primary manipulations in this exercise involves adjusting the input variable \( x \) to compress or translate the graph.

• **For Compression:** The function \( y = f(1.5x) \) adjusts each \( x \) by multiplying it by 1.5. An algebraic expression of this would show as \( 2 \sqrt{1.5x} - 2(1.5x) + 0.25(1.5x)^2 \). This reduces the span of the graph along the x-axis.
• **For Translation:** The transformation to \( y = f(x-1) + 0.5 \) involves moving \( x \) one unit to the right by modifying the equation to \( 2 \sqrt{x-1} - 2(x-1) + 0.25(x-1)^2 \) and then further lifting the function by adding 0.5.

Algebraic manipulation is therefore not only about solving equations but understanding how each change in the expression affects the graph. It's through this understanding that you develop a keen insight into the more abstract concept of function transformations resulting from these manipulations.