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Magazine Chapter 5: Problem 28
Sketch the line with the given slope and \(y\) -intercept. $$m=-4,(0,-2)$$
Short Answer
Expert verified
The line has a slope of -4 and passes through the point (0, -2).
Step by step solution
01
Understanding the Slope-Intercept Form
The equation of a line in slope-intercept form is given by \(y = mx + c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Here, the slope \(m = -4\) and the \(y\)-intercept is \(c = -2\). So, the equation of the line becomes \(y = -4x - 2\).
02
Identify the Y-Intercept on the Graph
The \(y\)-intercept is the point where the line crosses the \(y\)-axis. This occurs when \(x = 0\). From the given information, the \(y\)-intercept is \((0, -2)\). Plot this point on the graph.
03
Use the Slope to Determine Another Point
The slope \(m = -4\) implies that for every 1 unit increase in \(x\), \(y\) decreases by 4 units (since the slope is negative). Starting from the \(y\)-intercept \((0, -2)\), move 1 unit to the right (to \(x = 1\)) and 4 units down (to \(y = -6\)). This gives another point: \((1, -6)\). Plot this point on the graph as well.
04
Draw the Line Through the Points
Using a ruler, draw a straight line through the two points \((0, -2)\) and \((1, -6)\). Extend the line through these points across the graph to illustrate the path of the line represented by the equation \(y = -4x - 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equation
A linear equation is a mathematical statement that forms a straight line when graphed on a coordinate plane. The most common way to express a linear equation is by using the slope-intercept form: \( y = mx + c \). In this form, the variable \( y \) represents the dependent variable dependent on \( x \), the independent variable. The equation describes how \( y \) changes in response to \( x \). Linear equations are fundamental in understanding relationships where change between variables is consistent.
When dealing with linear equations, keep in mind these key ideas:
When dealing with linear equations, keep in mind these key ideas:
- The graph will always be a straight line.
- The equation highlights the rate of change in \( y \) with respect to \( x \), which is consistent.
- It is useful for predicting values and identifying trends when only some of the data is known.
Graphing a Line
Graphing a line involves plotting points and drawing a straight path that passes through them. This helps visualize the relationship between variables expressed in a linear equation. If you have an equation in the form \( y = mx + c \), you can start with the \( y \)-intercept and use the slope to locate another point.
Here鈥檚 how you typically graph a line:
This method helps you visualize how the slope and \( y \)-intercept interact on the graph to form a straight line.
Here鈥檚 how you typically graph a line:
- Begin by plotting the \( y \)-intercept on the graph, which is the point where the line crosses the \( y \)-axis.
- Next, use the slope to find another point. For example, if the slope is \(-4\), move 4 units down for every 1 unit you move to the right.
- Connect the plotted points using a straight edge to draw the line.
This method helps you visualize how the slope and \( y \)-intercept interact on the graph to form a straight line.
Y-Intercept
The \( y \)-intercept is where the graph of the equation touches the \( y \)-axis. It's represented by the point \((0, c)\) in the equation \( y = mx + c \). Here, \( c \) denotes the exact point on the \( y \)-axis where the line cuts through. The concept of a \( y \)-intercept is critical because it provides a starting point for graphing the line.
Understanding the \( y \)-intercept is key to quickly establishing your line graph's orientation.
- The \( y \)-intercept is easily found within the equation as the constant term, \( c \).
- This point acts as a stable reference when graphing a line since it doesn't change with different values of \( x \).
Understanding the \( y \)-intercept is key to quickly establishing your line graph's orientation.
Slope
The slope of a line is a measure of its steepness or incline, depicted in the slope-intercept form \( y = mx + c \) by the letter \( m \). It shows how much \( y \) changes for a given change in \( x \), and can be either positive, negative, zero, or undefined. The value of the slope helps determine the direction of the line.
For example:
The slope is thus essential for understanding the relationship between the variables in a linear equation and how to properly sketch the line on the graph.
For example:
- A positive slope means the line ascends from left to right.
- A negative slope, like \(-4\) as in the exercise, means the line descends as you move from left to right.
- A zero slope indicates a horizontal line, showing no change in \( y \) as \( x \) changes.
The slope is thus essential for understanding the relationship between the variables in a linear equation and how to properly sketch the line on the graph.
