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Chapter 1: Problem 20

Graph the function and its parent function. Then describe the transformation. \(g(x)=4 x\)

Short Answer

Expert verified
The function \(g(x) = 4x\) is obtained from its parent function \(f(x) = x\) by a vertical stretch by a factor of 4.

Step by step solution

01

Plot the Parent Function

Draw a graph of the parent function which is \(f(x) = x\). The graph will be a straight line that passes through the origin. Any point (a,a) is on this line because \(f(a) = a\).
02

Plot the Given Function

The given function is \(g(x) = 4x\). To plot this function, calculate the values of \(y\) for a certain range of \(x\). Notice that any \((a,4a)\) will be on the line of \(g(x) = 4x\). Plot these points and draw a line through them. This will result in a line that is steeper than the parent function.
03

Describe the Transformation

Compare the graph of \(g(x) = 4x\) with the parent function \(f(x) = x\). It is clearly seen that the line \(g(x) = 4x\) is steeper than the line \(f(x) = x\). In terms of transformations, this means that the function \(g\) is the result of a vertical stretch by a factor of 4 to the parent function \(f\). This is because the value of \(y\) in the function \(g(x)\) is 4 times the value of \(y\) in equivalent points in the function \(f(x)\).

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Key Concepts

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

Parent Function
The concept of a "parent function" is a foundational idea in understanding function transformations. A parent function is the simplest form of a function family that maintains the core characteristics of that family. In the case of linear functions, the parent function is typically denoted as \(f(x) = x\). This function creates a straight line that passes directly through the origin (0,0) and has a slope of 1. The beauty of the parent function lies in its simplicity:
  • The graph is a direct representation of a linear relationship where every increase in \(x\) results in the same increase in \(y\).
  • All points on this line take the form \((a, a)\), exemplifying a one-to-one relationship.
This makes the parent function an excellent starting point for exploring other more complex functions by applying transformations such as stretches, compressions, translations, or reflections.
Vertical Stretch
When discussing function transformations, a "vertical stretch" refers to a transformation that increases the distance between points on the graph and the x-axis, while maintaining the x-intercepts. In mathematical terms, a vertical stretch occurs when every output value, or \(y\)-value, of a function is multiplied by a constant factor greater than 1. For the function \(g(x) = 4x\), this factor is 4:
  • This means that for every point \((a, a)\) on the parent function \(f(x) = x\), there will be a corresponding point \((a, 4a)\) on \(g(x) = 4x\).
  • The line becomes steeper, accentuating the change in \(y\) values relative to the \(x\) values.
As the constant factor increases, the steepness of the line increases as well. Vertical stretches are a crucial way to introduce varying rates of change within linear functions, adding diversity to graphs while maintaining the original line's direction.
Linear Functions
Linear functions are one of the simplest forms of mathematical functions, characterized by their straight-line graph. These functions are generally expressed in the formula \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. Key attributes of linear functions include:
  • They have a constant rate of change, which means they increase or decrease consistently across their domain.
  • The slope \(m\) dictates the angle or steepness of the line, whereas the y-intercept \(b\) indicates where the line crosses the y-axis.
  • Linear functions are easy to interpret because they visually represent a direct and constant relationship between \(x\) and \(y\).
In the context of the exercise, \(g(x) = 4x\) is a linear function with a slope of 4 and a y-intercept of 0, highlighting its simple yet powerful form. Understanding this basic form allows students to apply transformations and comprehend more complex functions deeply.

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Most popular questions from this chapter

In Exercises 39-44, identify the function family and describe the domain and range. Use a graphing calculator to verify your answer. \(g(x)=|x+2|-1\)

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=|x+3| ;\) horizontal stretch by a factor of 4

Use a graphing calculator to graph the function and its parent function. Then describe the transformations. \(g(x)=-|x-1|-\frac{1}{2}\)

Find the values of \(a, b\), and \(c\) so that the linear system shown has \((-1,2,-3)\) as its only solution. Explain your reasoning. $$ \begin{aligned} &x+2 y-3 z=a \\ &-x-y+z=b \\ &2 x+3 y-2 z=c \end{aligned} $$

Simplify \((2 z-5)^2\)
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