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Magazine Chapter 1: Problem 2
Sketch, on the same set of axes, lines passing through point (1,3) and having slopes \(-3,-2,-1,0,1,2,\) and 3
Short Answer
Expert verified
Plot lines with equations: \( y = -3x + 6, y = -2x + 5, y = -x + 4, y = 3, y = x + 2, y = 2x + 1, y = 3x \).
Step by step solution
01
Understanding the Equation of a Line
The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For lines passing through a specific point, such as \((1, 3)\), we can determine \( b \) by substituting the point into the equation.
02
Substitute Point into Equation for Each Slope
The point \((1, 3)\) means \( x = 1 \) and \( y = 3 \). Substitute \( m = -3 \) into \( 3 = -3(1) + b \), which gives \( b = 6 \). Repeat for each slope: \( -2, -1, 0, 1, 2, \) and \( 3 \). The solutions for \( b \) are: - \(-2: b = 5\)- \(-1: b = 4\)- \( 0: b = 3\)- \( 1: b = 2\)- \( 2: b = 1\)- \( 3: b = 0\).
03
Write the Equation of Each Line
Using the values of \( m \) and the corresponding \( b \), write the equations for each line:- \( y = -3x + 6 \)- \( y = -2x + 5 \)- \( y = -x + 4 \)- \( y = 3 \)- \( y = x + 2 \)- \( y = 2x + 1 \)- \( y = 3x \).
04
Sketch the Lines on a Graph
On a set of axes, plot each of these lines starting from the y-intercept, using the slope to find the next points. For example, for \( y = -3x + 6 \), plot the point \( (0, 6) \) and move down 3 units and 1 unit right for each step. Repeat for each line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equation of a Line
The equation of a line is a fundamental concept in calculus and algebra that describes the relationship between two variables, typically represented as \( x \) and \( y \). This equation can take several forms, with the most common being the slope-intercept form and point-slope form. A line's equation allows us to understand its behavior in a coordinate plane.
The general form of a line's equation is \( y = mx + b \), where:
Understanding how to derive and use these parts is essential in graphing and analyzing lines in mathematics.
The general form of a line's equation is \( y = mx + b \), where:
- \( m \) represents the slope of the line, indicating its steepness and direction.
- \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
Understanding how to derive and use these parts is essential in graphing and analyzing lines in mathematics.
Slope-Intercept Form
The slope-intercept form is a straightforward and commonly used form of a linear equation expressed as \( y = mx + b \). It is particularly useful for quickly graphing a line and understanding its properties.
Here's why it's helpful:
Once you have both the slope and intercept, graphing becomes a straightforward process.
Here's why it's helpful:
- Easy to Plot: The y-intercept \( b \) provides a starting point on the graph, while the slope \( m \) tells you how the line moves from that point.
- Quick Analysis: By glancing at the equation, you can immediately tell if the line is increasing (positive slope), decreasing (negative slope), or horizontal (zero slope).
Once you have both the slope and intercept, graphing becomes a straightforward process.
Graphing Lines
Graphing lines is a crucial skill that allows you to visualize the relationship between variables. Once you have an equation in slope-intercept form, you can easily plot the line on a coordinate plane.
To graph a line:
Practicing with slopes and y-intercepts from given problems builds confidence in quickly sketching accurate graphs.
To graph a line:
- Start at the y-intercept, \( b \), plotting the point where the line crosses the y-axis.
- Use the slope \( m \) to determine the direction and steepness. Positive slopes rise to the right, while negative slopes fall.
- From the y-intercept, move vertically by the rise and horizontally by the run indicated by the slope to find another point. For instance, a slope of \(-3\) means dropping 3 units down for every 1 unit moved right.
Practicing with slopes and y-intercepts from given problems builds confidence in quickly sketching accurate graphs.
Point-Slope Calculation
Point-slope calculation is another useful method for finding the equation of a line, primarily when you have one point and the slope. The point-slope form is written as \( y - y_1 = m(x - x_1) \).
This method is beneficial when:
\[ y - 3 = -3(x - 1) \]
Solving for \( y \) converts it to slope-intercept form:
\[ y = -3x + 6 \]
Point-slope calculation offers a flexible approach when graphing or analyzing additional line features.
This method is beneficial when:
- You know a point \((x_1, y_1)\) on the line and the slope \( m \).
- You need to derive the line's equation without starting with the y-intercept.
\[ y - 3 = -3(x - 1) \]
Solving for \( y \) converts it to slope-intercept form:
\[ y = -3x + 6 \]
Point-slope calculation offers a flexible approach when graphing or analyzing additional line features.
