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Magazine Chapter 1: Problem 27
Graph each function. $$ f(x)=\left\\{\begin{array}{ll} 2 x-7 & \text { if } x \geq 4 \\ 2-x & \text { if } x<4 \end{array}\right. $$
Short Answer
Expert verified
Graph two lines: one starting at (4, 1) with a positive slope, and the other extending left of \( x = 4 \) with a negative slope.
Step by step solution
01
Understand the Function
The given function is a piecewise function composed of two separate expressions. For \( x \geq 4 \), the function is \( f(x) = 2x - 7 \). For \( x < 4 \), the function is \( f(x) = 2 - x \).
02
Analyze the First Piece
Review the first piece \( f(x) = 2x - 7 \), which applies for \( x \geq 4 \). This is a linear equation with a slope of 2 and a y-intercept of -7. We will focus this line starting from \( x = 4 \) to the right.
03
Plot Points for the First Piece
Calculate \( f(x) \) for \( x = 4 \). \( f(4) = 2(4) - 7 = 1 \). This point (4, 1) will be included on the graph. As \( x \) increases, calculate additional points if necessary to confirm direction and slope.
04
Analyze the Second Piece
The second piece is \( f(x) = 2 - x \), which applies for \( x < 4 \). This is a linear equation with an inverted slope of -1 and a y-intercept of 2. This part of the graph will extend to the left of \( x = 4 \).
05
Plot Points for the Second Piece
Calculate \( f(x) \) for values less than 4. At \( x = 3 \), \( f(3) = 2 - 3 = -1 \). Calculate more points as needed. The line continues leftward as \( x \) decreases.
06
Connect and Graph
Using the calculated points, draw the line segments according to each piece. The first line segment should start at (4, 1) and continue with a positive slope. The second line segment starts left from just before \( x = 4 \) and continues with a negative slope. Check continuity and endpoints for clarity in graphing.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
A linear equation is a fundamental concept in mathematics, which represents a straight line on a graph. The standard form of a linear equation in two variables is usually written as \( y = mx + b \). In this form, \( m \) represents the slope, which indicates how steep the line is, and \( b \) represents the y-intercept, which is where the line crosses the y-axis.
In the context of our piecewise function, each part of the function is a linear equation:
In the context of our piecewise function, each part of the function is a linear equation:
- For \( x \geq 4 \), the equation is \( f(x) = 2x - 7 \). Here, the slope \( m = 2 \) means the line rises 2 units for every 1 unit it moves to the right. The y-intercept \(-7\) tells us that this line crosses the y-axis at \( y = -7 \).
- For \( x < 4 \), the equation is \( f(x) = 2 - x \), which can also be written as \( f(x) = -x + 2 \). In this case, the slope \( m = -1 \) indicates that the line falls 1 unit for every 1 unit it moves to the right, and the y-intercept \( b = 2 \) means it crosses the y-axis at \( y = 2 \).
Graphing Functions
To successfully graph a piecewise function, it's essential to plot each segment of the function accurately by considering the rules that define their domains. Following these steps will aid in visualizing the function thoroughly:
First, identify each piece's equation and its corresponding domain. For \( f(x) \), you have two pieces:
First, identify each piece's equation and its corresponding domain. For \( f(x) \), you have two pieces:
- \( f(x) = 2x - 7 \) for \( x \geq 4 \)
- \( f(x) = 2 - x \) for \( x < 4 \)
- For \( x = 4 \), \( f(4) = 2(4) - 7 = 1 \). Therefore, the point (4, 1) marks the beginning of the \( 2x - 7 \) portion.
- Another point on this line could be found by choosing \( x = 5 \), leading to \( f(5) = 2(5) - 7 = 3 \).
- For \( x = 3 \), \( f(3) = 2 - 3 = -1 \), and you can select further points to ensure accuracy.
Function Analysis
Analyzing piecewise functions involves dissecting how each part behaves within its specific interval. To thoroughly understand this, you should:
- Identify intervals: Here, \( f(x) \) changes its formula at \( x = 4 \).
- Check for continuity: The function transitions from one rule to another at the breaking point, \( x = 4 \). Evaluating: \( f(4) = 1 \) for both pieces ensures they connect without gaps or overlaps.
- Determine slope impact: A slope of 2 in the interval \( x \geq 4 \) creates a rising line, while a slope of -1 for \( x < 4 \) causes a descending line. The distinction in slopes illustrates changes in steepness between parts.
- Analyze endpoints: Examine if the endpoints of intervals align correctly; in this exercise, both parts transition smoothly at \( x = 4 \) without jumps.
