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Chapter 7: Problem 17

Graph each equation using the slope and \(y\) -intercept. $$y=2 x-3$$

Short Answer

Expert verified
Plot the y-intercept at (0, -3) and use slope 2 to plot point (1, -1); connect the points with a line.

Step by step solution

01

Identify the slope and y-intercept

In the equation \(y = 2x - 3\), the slope \(m\) is 2 and the \(y\)-intercept \(b\) is -3. The slope tells us that for every increase of 1 in \(x\), \(y\) increases by 2. The \(y\)-intercept tells where the line crosses the \(y\)-axis.
02

Plot the y-intercept

Start by plotting the \(y\)-intercept (0, -3) on the graph. This point is where the line will intersect the \(y\)-axis.
03

Use the slope to find another point

From the \(y\)-intercept (0, -3), use the slope to find another point on the line. Since the slope is 2, move up 2 units and right 1 unit to reach the point (1, -1). Plot this point on the graph.
04

Draw the line

Connect the two points, (0, -3) and (1, -1), with a straight line. Extend the line in both directions, adding arrowheads to indicate that it continues indefinitely.

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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 of a linear equation makes graphing much simpler. This form is written as \( y = mx + b \). In it, \( m \) represents the slope of the line, and \( b \) represents the \( y \)-intercept. This format allows you to quickly understand how steep a line is and where it crosses the \( y \)-axis. Knowing these two components is crucial as it provides a direct insight into the behavior of the line on a graph without needing to create a table of values. Remember, the higher the absolute value of the slope, the steeper the line becomes. Meanwhile, the \( y \)-intercept directly corresponds to a specific point on the graph, making it an anchor for plotting.
Plotting Points
Plotting points is an essential skill for visualizing equations graphically. Once you have your slope and \( y \)-intercept from a linear equation, you can start plotting. Begin by plotting the \( y \)-intercept point, which gives you a solid starting location. From there, use the slope to determine the next point.
  • Mark the \( y \)-intercept on the graph.
  • From the \( y \)-intercept, use the slope to find another point.
This process allows us to visualize the line's trajectory clearly. It鈥檚 more than just placing points, as it accurately reflects the equation described. This method helps in seeing the direction and angle at which the line progresses.
Slope of a Line
The slope of a line is a measure of its steepness, indicating how much the line rises or falls as it moves horizontally across the graph. In the slope-intercept form \( y = mx + b \), the slope is represented by \( m \).
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope indicates the line falls as it moves from left to right.
  • If the slope is zero, the line is perfectly horizontal.
  • A vertical line has an undefined slope.
In our equation \( y = 2x - 3 \), a slope \( m \) of 2 means that for every 1 unit you move to the right along the \( x \)-axis, the line will move up 2 units. This is a visual representation of the rate of change - showing the relationship between the change in \( y \) and the change in \( x \).
y-Intercept
The \( y \)-intercept is a crucial element when graphing linear equations. It is the point where the line crosses the \( y \)-axis. In the equation \( y = mx + b \), the \( y \)-intercept is denoted by \( b \). This value gives the starting point for the graph in the vertical direction.
  • In our example \( y = 2x - 3 \), the \( y \)-intercept \( b \) is -3, meaning the line crosses the \( y \)-axis at (0, -3).
  • The intercept shows the initial value of \( y \) when \( x \) is zero.
This initial crossing point helps establish the line's foundation on a graph, which, when combined with another point derived using the slope, allows the full line to be drawn. Remember, the \( y \)-intercept is always plotted before extending the line using the slope.

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Most popular questions from this chapter

How can linear equations represent a function? Include in your answer a description of four ways that you can represent a function, and an example of a linear equation that could be used to determine the cost of \(x\) pounds of bananas that are 0.49 dollars per pound.

Evaluate each expression. (Lesson \(1-2\) ) $$\frac{46-22}{2005-2001}$$

Simplify. $$2(18)-1$$

State the slope and the \(y\) -intercept of the graph of each equation. $$y=x+5$$

Solve each equation for \(y .\) $$-x+5 y=10$$
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