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Magazine Chapter 2: Problem 31
Graph each equation by hand. $$y=1-x, y=|1-x|$$
Short Answer
Expert verified
Graph \( y = 1-x \) as a straight line with negative slope; \( y = |1-x| \) as a V-shape intersecting at (1,0).
Step by step solution
01
Understand Linear Equation
The linear equation given is \( y = 1 - x \). This is a straight line with a y-intercept of 1 and a slope of -1. This means for every 1 unit increase in \( x \), \( y \) decreases by 1.
02
Draw Linear Equation
To graph \( y = 1 - x \), start at the point (0, 1) on the y-axis, which is the y-intercept. From this point, use the slope (-1) to find another point: move 1 unit right (to x = 1) and 1 unit down (to y = 0), giving the point (1, 0). Plot these points and draw a straight line through them.
03
Understand Absolute Value Equation
The equation \( y = |1-x| \) is similar to the linear equation but reflects all negative parts of \( y \) to positive in the graph. This changes negative slopes into reflections, creating a V-shape.
04
Determine Critical Points for Absolute Value
The critical point is where \( 1-x = 0 \) or \( x = 1 \). Here, \( y = 0 \). For points with \( x < 1 \), \( 1-x > 0 \), so \( y \) follows \( y = 1-x \). For \( x > 1 \), \( 1-x < 0 \), so \( y = -(1-x) = x-1 \). The critical point is (1, 0).
05
Draw Absolute Value Equation
Starting from the origin of each segment, from x-axis intercept (0, 1) for \( x < 1 \), graph \( y = 1-x \). Reflected about the line \( x=1 \) for \( x > 1 \), graph \( y = x-1 \). Combine these segments to form a V-shape that opens towards negative y-axis but mirrors upwards after x=1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
Linear equations are equations that form a straight line when graphed on a coordinate plane. They are usually written in the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. In the case of the exercise given, the linear equation is \( y = 1 - x \).
- Y-intercept: This is the point where the line crosses the y-axis. For the equation \( y = 1 - x \), the y-intercept is at point (0, 1).
- Slope: The slope of a line indicates its steepness and direction. In this equation, the slope is \(-1\). This means that for every increase of 1 unit in \( x \), \( y \) decreases by 1 unit.
Absolute Value Functions
Absolute value functions are known for their iconic V-shaped graphs. The absolute value of a number is its distance from zero on the number line, regardless of direction. The equation given is \( y = |1 - x| \). This signifies taking the absolute value of the linear expression \( 1 - x \).
- Positive Output: Absolute value functions convert any negative output to positive. Thus, when \( 1 - x \) is negative, the graph flips the value to positive, creating a reflection.
- Graph Shape: The V-shape is the result of reflecting all parts of the line that would fall below the x-axis upwards.
- Critical Point: This is where the absolute value "turns," and it can be found where the expression inside bars equals zero. Here, it occurs at \( x = 1 \).
Critical Points
In graphing, critical points are key to understanding where changes or important features occur, like peaks, valleys, or intercepts. For the absolute value function \( y = |1 - x| \), the critical point is where the expression inside the absolute value equals zero.
- Finding Critical Points: Set \( 1 - x = 0 \) to find that \( x = 1 \) is the critical point.
- Physically on Graph: This point is (1, 0), where the graph of the absolute value function transitions or "turns" at the vertex of the V-shape.
Graphing Transformations
Graphing transformations involve shifting, reflecting, stretching, or compressing graphs of functions. For the absolute value function \( y = |1 - x| \), we can regard it as being derived from the linear function \( y = 1 - x \) by way of reflection.
- Reflection: The primary transformation here is reflection. Negative values that would normally appear below the x-axis are reflected above it, modifying the linear graph into a V-shape.
- Transformation Effects: These transformations are used to adjust the graph to match the functional requirements, which in this case, means no negative y-values.
