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Chapter 1: Problem 5

Identify the function family to which \(f\) belongs. Compare the graph of \(f\) to the graph of its parent function. \(f(x)=5 x-2\)

Short Answer

Expert verified
The function \(f(x) = 5x - 2\) belongs to the family of linear functions. Its parent function is \(f(x) = x\). The graph of \(f(x) = 5x - 2\) is a steeper, downward-shifted version of the graph of \(f(x) = x\).

Step by step solution

01

Identify the Function Family

By looking at the equation \(f(x) = 5x - 2\), we can identify it as a linear function. In general, any function that can be written in the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants, is a linear function.
02

Identify the Parent Function

The parent function of a family of functions is the simplest form of that function. In the case of linear functions, the parent function is \(f(x) = x\).
03

Compare the Graphs

Now, comparing the graph of the given function with the graph of the parent function, we can notice that there is a vertical shift of 2 units down and a steeper slope in the given function, \(f(x) = 5x - 2\). The slope of the line is five, which means it is steeper than the parent function \(f(x) = x\), and the '-2' in the equation indicates a vertical shift down by 2 units. The graph of \(f(x) = 5x - 2\) is therefore a steeper, downward-shifted version of the graph of \(f(x) = x\).

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Key Concepts

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

Parent Function
In mathematics, the parent function serves as the most basic form of a function family. It provides a starting point for analyzing other functions in that family. For linear functions, the parent function is simple:
  • The equation is written as \(f(x) = x\).
This equation represents a straight line that passes through the origin, with a 1:1 ratio between x and y, producing a line with a slope of 1. Understanding the parent function helps in recognizing how other functions within the same family are derived or transformed. Every transformation of the parent function, like shifts or changes in steepness, allows us to visualize and interpret more complex structures.
Vertical Shift
A vertical shift occurs when a graph moves up or down from its original position. In function notation, this is represented by adding or subtracting a constant from the function鈥檚 formula. For example:
  • In \(f(x) = 5x - 2\), the '-2' represents a vertical shift.
This specific vertical shift moves the graph 2 units down. Consequently, every point on the graph is lowered by the same amount. Vertical shifts are important because they change the graph's y-intercept without affecting its slope. It allows the function to intersect a different part of the y-axis, making altered lines still parallel to the parent function unless other transformations are applied.
Slope
The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the change in y to the change in x, often referred to as "rise over run." In the equation \(f(x) = mx + b\):
  • 'm' represents the slope.
For the equation \(f(x) = 5x -2\), the slope is 5, indicating that for every one unit the function moves horizontally, it moves five units vertically. This steepness means the line rises sharply compared to the parent function, which has a slope of 1. The slope not only affects the angle of the line but also conveys how rapidly the dependent variable changes with respect to the independent variable.
Graph Comparison
Comparing the graphs of two functions allows us to see the effects of different transformations clearly. When we compare \(f(x) = 5x - 2\) to its parent function \(f(x) = x\), we notice two main differences:
  • The shift involves the graph moving 2 units downward, creating a new y-intercept for the function.
  • The slope of 5 increases the line's steepness, causing it to rise more sharply than the parent function.
These differences highlight the impact of changing the slope and applying a vertical shift. By examining the transformations visually, we appreciate the manner in which alterations to the function equation translate into changes in the function鈥檚 graph. Understanding these conversions is essential for solving more complex problems involving linear equations and their graphs.

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Most popular questions from this chapter

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