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Magazine Chapter 2: Problem 37
Graph each equation by hand. $$y=-2 x-4, y=|-2 x-4|$$
Short Answer
Expert verified
The graphs are a line \( y = -2x - 4 \) and a V-shape for \( y = |-2x - 4| \).
Step by step solution
01
Identify the Linear Equation
The first equation we need to graph is a linear equation: \( y = -2x - 4 \). This equation is in the slope-intercept form \( y = mx + b \), where the slope \( m = -2 \) and the y-intercept \( b = -4 \).
02
Graph the Linear Equation
To graph \( y = -2x - 4 \), start by plotting the y-intercept at \( (0, -4) \) on the coordinate plane. Use the slope \(-2\), which means for every 1 unit you move to the right (positive direction on x-axis), move 2 units down (negative direction on y-axis). From \( (0, -4) \), move to \( (1, -6) \). Continue this pattern to plot a few points and draw a straight line through these points.
03
Identify the Absolute Value Equation
The second equation is \( y = |-2x - 4| \). This represents the absolute value of the linear equation. The absolute value affects how the line will appear on the graph: all negative y-values become positive.
04
Understand the Effect of Absolute Value
The effect of the absolute value is to 'reflect' all portions of the graph of \( y = -2x - 4 \) that are below the x-axis (where \( y < 0 \)) upward, above the x-axis. This creates a V-shape.
05
Graph the Absolute Value Equation
Start with the graph of \( y = -2x - 4 \). Observe where this graph intersects the x-axis. Solve \( -2x - 4 = 0 \) to find the x-coordinate of this intersection point. Solve to get \( x = -2 \), thus \( (-2, 0) \) is where the line crosses the x-axis. For points to the right of this intersection, reflect the linear plot above the x-axis to create the V-shape of \( y = |-2x -4| \). For x-values left of \( -2 \), the graph is simply the reflection of \( y = -2x - 4 \) above the x-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form is one of the most common ways to express a linear equation. It is shown as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This format makes it easy to quickly graph linear equations. Here's why it's useful:
This format makes it easy to quickly graph linear equations. Here's why it's useful:
- Slope (\(m\)): This tells us how steep the line is. It represents the change in \(y\) when \(x\) increases by one unit. A positive slope means the line rises as you move to the right, while a negative slope means the line falls.
- Y-Intercept (\(b\)): This is the point where the line crosses the y-axis. It gives a starting point to draw the line.
Linear Equations
Linear equations are fundamental in algebra and they represent straight lines when graphed on a coordinate plane.
They take the general form \(y = mx + b\), but can also appear in different forms, like standard or point-slope.
Here鈥檚 what you need to know about linear equations:
They take the general form \(y = mx + b\), but can also appear in different forms, like standard or point-slope.
Here鈥檚 what you need to know about linear equations:
- Characteristics: They graph as straight lines and have a constant slope.
- Graphing: Start with the y-intercept and use the slope to find additional points.
- Solutions: Any point (\(x, y\)) on the line is a solution to the equation.
Absolute Value Transformations
Absolute value transformations change how a function appears on a graph, particularly by modifying its shape.
For the function \(y = |f(x)|\), any negative \(y\)-values of the function \(f(x)\) become positive.
Let's explore what this means:
For the function \(y = |f(x)|\), any negative \(y\)-values of the function \(f(x)\) become positive.
Let's explore what this means:
- Reflection: Negative outputs of the function are reflected across the x-axis. This creates a "V" shape, especially noticeable in simple linear equations like \(y = |m x + b|\).
- Vertex: The point on the graph where the direction changes is known as the vertex, typically a point of reflection.
- Examples: For the equation \(y = |-2x - 4|\), after graphing \(-2x - 4\) normally, you simply "flip" the graph at the x-axis to ensure all y-values are non-negative.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we graph equations and is crucial for visualizing mathematical concepts.
It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Here is more to it:
It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Here is more to it:
- Quadrants: The plane is divided into four quadrants. Each quadrant sign is determined by the positive or negative values of x and y.
- Points: Any point on the plane is represented by a pair of numbers (\(x, y\)), indicating its horizontal and vertical positions.
- Graphing Functions: By plotting different points that an equation satisfies, lines or curves take shape on this plane.
