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Magazine Chapter 3: Problem 69
Graph the indicated functions. Plot the graphs of (a) \(y=x+2\) and (b) \(y=\frac{x^{2}-4}{x-2}\) Explain the difference between the graphs.
Short Answer
Expert verified
\(y=x+2\) is a straight line; \(y=\frac{x^2-4}{x-2}\) is the same line but with a hole at \(x=2\).
Step by step solution
01
Understand the function (a)
The function given is a linear equation in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( y = x + 2 \) means the slope \( m = 1 \) and the y-intercept \( b = 2 \). This indicates a straight line.
02
Graph the linear function (a)
To graph \( y = x + 2 \), start by identifying the y-intercept at point (0, 2) on the graph. From this point, use the slope of 1 to plot another point by going up 1 unit and right 1 unit. Draw a straight line connecting these points to extend the graph.
03
Understand the function (b)
The function \( y = \frac{x^2-4}{x-2} \) is a rational function. First, factor the numerator: \( x^2 - 4 = (x-2)(x+2) \). Therefore, the function becomes \( y = \frac{(x-2)(x+2)}{x-2} \), which reduces to \( y = x+2 \) for \( x eq 2 \). However, at \( x = 2 \), the function is undefined.
04
Graph the rational function (b)
To graph \( y = \frac{x^2-4}{x-2} \), plot the line \( y = x+2 \) as you did in step 2, but place an open circle at \( x = 2 \) to indicate the point is undefined. This creates a point of discontinuity, an indication that the function is not defined at that specific x-value.
05
Compare the graphs
The graph of \( y = x+2 \) is a straight line without interruptions, while \( y = \frac{x^2-4}{x-2} \) appears the same except for an open circle at \( x = 2 \), indicating the function is undefined at this point. This is due to the factor \( (x-2) \) being present in both the numerator and denominator, creating a removable discontinuity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Functions
A linear function is a type of function where the graph forms a straight line. The standard form of a linear function is written as \( y = mx + b \). Here, \( m \) is known as the "slope," and \( b \) is the "y-intercept." This format is particularly user-friendly because both the slope and y-intercept offer direct, actionable insights into graphing the line. Linear functions are very predictable in their graphical representation, as they do not curve or change direction.
- Slope (\(m\)): This value indicates how steep the line is. A positive slope means the line inclines upward, while a negative slope means it declines downward.
- Y-intercept (\(b\)): This is where the line crosses the y-axis. Knowing this point is essential for plotting the function quickly.
Rational Functions
Rational functions are a bit more complex. They consist of the ratio of two polynomial functions. For example, the function \( y = \frac{x^2 - 4}{x - 2} \) is a rational function. Identifying and understanding these can be a bit tricky because they are not always defined across every real number.
- The numerator is \(x^2 - 4\), which can be factored into \((x-2)(x+2)\).
- The denominator is \(x-2\). However, this causes the function to be undefined when \(x = 2\), as you can't divide by zero.
Discontinuity
Discontinuity is a key concept to understand while dealing with functions, especially rational functions. When a function has a point of discontinuity, it is not defined at that specific point. In our example, the function \( y = \frac{x^2-4}{x-2} \) becomes undefined at \( x = 2 \) because substituting this value into the denominator results in a division by zero.
- Removable Discontinuity: Seen in functions like ours, this occurs when both the numerator and denominator share a common factor, here \(x-2\).
- The graph of the function is missing just one point, indicated by an open circle at \(x = 2\).
Slope-Intercept Form
The slope-intercept form of a linear function is expressed as \( y = mx + b \). This form is widely used because it provides immediate information on how to graph the equation.
The slope \(m\) tells us how the y-value changes with respect to x-value. For example, if \( m = 1 \), for every one-unit increase in x, y increases by one unit. The y-intercept \(b\) indicates where the line meets the y-axis. In the function \( y = x + 2 \), the y-intercept is 2, meaning the line crosses the y-axis at (0, 2).
The slope \(m\) tells us how the y-value changes with respect to x-value. For example, if \( m = 1 \), for every one-unit increase in x, y increases by one unit. The y-intercept \(b\) indicates where the line meets the y-axis. In the function \( y = x + 2 \), the y-intercept is 2, meaning the line crosses the y-axis at (0, 2).
- The slope and y-intercept make it very straightforward to draw the graph.
- By understanding the slope-intercept form, we can quickly determine key points through which the line will pass.
