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Magazine Chapter 1: Problem 55
Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? \(y=m(x-3) \quad\) for \(m=0, \pm 0.25, \pm 0.75, \pm 1.5\)
Short Answer
Expert verified
All lines intersect at the point \((3, 0)\).
Step by step solution
01
Understanding the Equation
The general equation given is in the form \(y = m(x - 3)\). This represents a family of lines where each line passes through a point that shifts horizontally with varying slopes \(m\). The constant in \(x - 3\) shows the horizontal translation 3 units to the right on the x-axis.
02
Identifying Slope Values
The problem provides different slope values \(m = 0, \pm 0.25, \pm 0.75, \pm 1.5\). Each of these values will give a line with a different steepness. When \(m = 0\), the line is horizontal. Positive values of \(m\) indicate lines rising to the right, while negative values indicate lines falling to the right.
03
Graphing Each Line
Use a graphing device or software to plot each line \(y = m(x - 3)\) for each given \(m\) value. Here's how each line will look:- \(m = 0: y = 0 \times (x - 3) \Rightarrow y = 0\). This is a horizontal line at \(y = 0\).- \(m = 0.25: y = 0.25(x - 3)\).- \(m = -0.25: y = -0.25(x - 3)\).- \(m = 0.75: y = 0.75(x - 3)\).- \(m = -0.75: y = -0.75(x - 3)\).- \(m = 1.5: y = 1.5(x - 3)\).- \(m = -1.5: y = -1.5(x - 3)\).
04
Analyzing the Graph
Observe the graph with all plotted lines. Notice where all lines intersect. Since each equation simplifies to a line through \((3, 0)\), all lines in this family intersect at the point \(x = 3\). This is because \(y = m(x - 3)\) forces each line to pass through this point, regardless of the slope \(m\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope of a Line
The slope of a line is a measure of its steepness or inclination. In the equation \(y = m(x - 3)\), the variable \(m\) represents the slope. A quick way to visualize this is to think about how the line rises or falls as you move from left to right.
Here are some key points about slope:
Here are some key points about slope:
- If \(m = 0\), the line is flat, meaning there is no rise, thus a horizontal line.
- Positive slopes \(m > 0\) result in lines that climb upward to the right.
- Negative slopes \(m < 0\) produce lines that fall downward to the right.
- The larger the absolute value of \(m\), the steeper the line.
Horizontal Translation
Horizontal translation involves shifting a graph left or right on the coordinate plane. In our equation \(y = m(x - 3)\), the expression \(x - 3\) indicates that every line in this set is shifted 3 units to the right along the x-axis.
It's important to note:
It's important to note:
- The horizontal shift does not affect the slope \(m\) of the lines.
- All lines will still move parallel to their original position but start from a new x-coordinate.
- Regardless of the slope, every line in the family passes through the point \((3, 0)\) as the common intersection due to this translation.
Intercepts
Intercepts are the points where a line crosses the x- or y-axis. For the family of lines given by \(y = m(x - 3)\), it is essential to identify these intercepts to understand their graphical behavior fully.
Here is how intercepts work for these equations:
Here is how intercepts work for these equations:
- Y-intercept: The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) in \(y = m(x - 3)\) gives \(y = -3m\). This tells us that each line crosses the y-axis at a different point depending on the slope \(m\).
- X-intercept: It's the point \((3, 0)\) shared among all equations in this particular family, which is due to the horizontal translation adjustment \(m(x - 3)\).
Family of Lines
A family of lines consists of multiple lines that share common characteristics. In our case, this family is described by \(y = m(x - 3)\), where the lines are related through variations in \(m\). Each line shows different behaviors based on its slope \(m\) but shares the same x-intercept at \(x = 3\).
Key points to understand include:
Key points to understand include:
- Lines are graphically distinguished by their slope; positive slopes rise, negative slopes fall, and a zero slope results in a horizontal line.
- All lines in this family intersect the x-axis at \((3, 0)\) due to the translation component in the equation.
- The lines do not intersect or overlap each other at any other points, showing parallel behavior when slopes are equal.
