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Chapter 2: Problem 36

Graphing Transformations Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$f(x)=\sqrt{x-4}$$

Short Answer

Expert verified
Shift \( g(x) = \sqrt{x} \) 4 units to the right to get \( f(x) = \sqrt{x-4} \).

Step by step solution

01

Identify the Standard Function

The given function is \( f(x) = \sqrt{x-4} \). The standard function here is \( g(x) = \sqrt{x} \), which is a square root function.
02

Horizontal Shift

The function \( f(x) = \sqrt{x-4} \) represents a horizontal shift of the standard function \( g(x) = \sqrt{x} \). The expression \( x-4 \) inside the square root indicates a shift to the right by 4 units. This is because replacing \( x \) with \( x-4 \) shifts the graph of the function \( g(x) \) rightward by the offset.
03

Sketch the Transformed Graph

Start with the graph of \( g(x) = \sqrt{x} \), which has its starting point (or vertex) at the origin (0,0), and has a characteristic curve that opens to the right. Shift the entire graph 4 units to the right. This means the new starting point is (4,0), and the graph keeps the same shape, extending to the right from this new starting point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function is one of the most fundamental functions encountered in pre-calculus and calculus. It is represented as \( g(x) = \sqrt{x} \). This function is only defined for non-negative values of \( x \), since the square root of a negative number is not a real number. The graph of the square root function has a starting point, also called the vertex, at the origin (0,0). From the origin, the graph opens to the right and moves upwards in a gentle curve.
  • The shape of the graph is characteristic, resembling half of a sideways parabola.
  • As \( x \) increases, the rate of increase of the square root function decreases, meaning the graph becomes less steep.
Understanding the square root function is crucial because it serves as the basis for more complex transformations in function graphs.
Horizontal Shift
A horizontal shift is one of the basic transformations that can be applied to a function's graph. In the expression \( f(x) = \sqrt{x-4} \), we identify a horizontal shift because of the term \( x-4 \) inside the square root. This expression indicates a shift to the right by 4 units.This transformation does not affect the shape of the square root function; it only changes its position on the coordinate plane.
  • The vertex of the function \( g(x) = \sqrt{x} \) is originally at (0,0).
  • After the transformation to \( f(x) = \sqrt{x-4} \), the vertex is moved to (4,0).
  • The entire graph is shifted to the right by the same amount, preserving its shape.
This simple horizontal shift helps in learning how to manipulate graphs, enabling the creation of complex graphical representations based on a standard function.
Standard Function
In mathematics, a standard function is a simple or basic function that other, more complex functions are often compared to, or derived from through transformations. The square root function \( g(x) = \sqrt{x} \) serves as a standard function in this context.Using a standard function as a starting point allows a precise understanding of how transformations affect a graph's structure and position.
  • Standard functions form the core templates for graphing transformations.
  • These are typically functions you start graphing without transformations.
  • Understanding these enables easier manipulation when adding shifts, stretches, or reflections.
When you graph a function like \( f(x) = \sqrt{x-4} \), you first recognize \( g(x) = \sqrt{x} \) as your standard function. This recognition simplifies plotting, as you already know the basic shape and behavior of the graph before applying any transformations.

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Most popular questions from this chapter

For every integer \(n\), the graph of the equation \(y=x^{n}\) is the graph of a function, namely \(f(x)=x^{n} .\) Explain why the graph of \(x=y^{2}\) is not the graph of a function of \(x .\) Is the graph of \(x=y^{3}\) the graph of a function of \(x ?\) If so, of what function of \(x\) is it the graph? Determine for what integers \(n\) the graph of \(x=y^{n}\) is a graph of a function of \(x\)

Graphing Functions Sketch a graph of the function by first making a table of values. $$k(x)=\sqrt[3]{-x}$$

When a bowl of hot soup is left in a room, the soup eventually cools down to room temperature. The temperature \(T\) of the soup is a function of time \(t .\) The table below gives the temperature (in "F) of a bowl of soup \(t\) minutes after it was set on the table. Find the average rate of change of the temperature of the soup over the first 20 minutes and over the next 20 minutes. During which interval did the soup cool off more quickly? $$\begin{array}{|c|c||c|c|} \hline t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) & t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) \\ \hline 0 & 200 & 35 & 94 \\ 5 & 172 & 40 & 89 \\ 10 & 150 & 50 & 81 \\ 15 & 133 & 60 & 77 \\ 20 & 119 & 90 & 72 \\ 25 & 108 & 120 & 70 \\ 30 & 100 & 150 & 70 \\ \hline \end{array}$$

A home owner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period beginning on a Sunday.

The table gives the population in a small coastal community for the period 1997-2006. Figures shown are for January 1 in each year. (a) What was the average rate of change of population between 1998 and 2001 ? (b) What was the average rate of change of population between 2002 and 2004 ? (c) For what period of time was the population increasing? (d) For what period of time was the population decreasing? $$\begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1997 & 624 \\ 1998 & 856 \\ 1999 & 1,336 \\ 2000 & 1,578 \\ 2001 & 1,591 \\ 2002 & 1,483 \\ 2003 & 994 \\ 2004 & 826 \\ 2005 & 801 \\ 2006 & 745 \\ \hline \end{array}$$
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