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Chapter 3: Problem 33

Graph each equation by constructing a table of values and then plotting the points. SEE EXAMPLE 2 . (OBJECTIVE 3) $$y=2 x-1$$

Short Answer

Expert verified
The graph of the equation \(y = 2x - 1\) is a straight line that passes through the points (-2, -5), (-1, -3), (0, -1), (1, 1), and (2, 3).

Step by step solution

01

Determine Some X-Values

To construct the table of values, first, choose some x-values to use. These can be any real numbers, but it's usually most convenient to use simple integers. To start with, let's choose -2, -1, 0, 1, 2.
02

Find Corresponding Y-Values

Plug each of the chosen x-values into the equation \(y = 2x - 1\). This gives the corresponding y-values:\n\nFor \(x = -2\), \(y = 2(-2) - 1 = -5\)\nFor \(x = -1\), \(y = 2(-1) - 1 = -3\)\nFor \(x = 0\), \(y = 2(0) - 1 = -1\)\nFor \(x = 1\), \(y = 2(1) - 1 = 1\)\nFor \(x = 2\), \(y = 2(2) - 1 = 3\)
03

Construct the Table of Values

Based on the x and y values obtained we can construct a table of values which looks like:\n\n\[\begin{array}{|c |c|}\hline\text{x} & \text{y} \\hline\text{-2} & \text{-5} \\text{-1} & \text{-3} \\text{0} & \text{-1} \\text{1} & \text{1} \\text{2} & \text{3} \\hline\end{array}\]
04

Plot the Points and Draw the Line

Plot the points (-2, -5), (-1, -3), (0, -1), (1, 1), and (2, 3) on the graph. Then draw a line passing through all of these points. This is the graph of the equation \(y = 2x - 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Table of Values
A table of values is a simple tool that helps you organize the coordinates you want to use when graphing a linear equation. By plugging specific x-values into the equation, you find corresponding y-values. This will result in pairs of numbers, known as ordered pairs, that you can later plot on a graph.

**Creating a Table of Values is Easy:**
  • Pick a few x-values. Generally, integers such as -2, -1, 0, 1, 2 are handy.
  • Substitute these x-values into the linear equation to find the y-values.
  • Arrange these x and y pairs in a table for easy reference.
In our example, for each x we found a corresponding y using the equation \( y = 2x - 1 \). The table then becomes a ready reference that guides the plotting process on a graph.
Plotting Points
The next step after constructing a table of values is to plot the points on a graph. Every pair of numbers from your table corresponds to a point on the graph, represented by an ordered pair (x, y). Placing these points accurately helps in drawing the correct graph of the equation.

**Procedure for Plotting Points:**
  • Read the ordered pairs from the table of values.
  • Start with the x-coordinate: move along the x-axis (horizontal line) to reach the correct position.
  • Then, use the y-coordinate: move up or down from this position along the y-axis (vertical line).
  • Mark the spot where your movements intersect 鈥� that's your point!
By accurately plotting the points (-2, -5), (-1, -3), (0, -1), (1, 1), and (2, 3), you visually depict the relationship defined by the equation \( y = 2x - 1 \). The precision of this process leads to a neatly drawn line that represents the equation.
Coordinate Plane
A coordinate plane is a two-dimensional surface on which we can graph lines and curves. It's made up of two number lines - the horizontal x-axis and the vertical y-axis. These axes divide the plane into four sections, known as quadrants, and intersect at the origin (0,0).

**Understanding the Coordinate Plane:**
  • The origin is your starting point for all graphing activities.
  • X-coordinates tell you how far to move left or right from the origin.
  • Y-coordinates tell you how far to move up or down from the origin.
  • Each point on the plane is defined by an (x, y) pair.
When graphing, always start from the origin and use the x- and y-coordinates to locate points. This structured approach ensures correctness when drawing graphs, such as the line for the equation \( y = 2x - 1 \).
Linear Function
A linear function is one of the simplest types of functions you can graph on a coordinate plane. It is represented by equations that look like \( y = mx + b \), where \( m \) and \( b \) are constants. The graph of a linear function is always a straight line.

**Facts about Linear Functions:**
  • The slope \( m \) affects the steepness and direction of the line.
  • The y-intercept \( b \) is where the line crosses the y-axis.
  • The equation \( y = 2x - 1 \) is a linear function because it can be rearranged into the form \( y = mx + b \).
  • To graph, you only need two points, but plotting more points improves accuracy.
Linear functions are foundational in algebra and help you understand more complex mathematical concepts. The straight-line graph resulting from the equation \( y = 2x - 1 \) is a perfect example.
Slope
The slope of a line is a measure of its steepness. In the equation of a linear function \( y = mx + b \), the slope is represented by \( m \). It tells you how much the y-value changes for a given change in the x-value.

**Key Points about Slope:**
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls.
  • A slope of zero indicates a horizontal line.
  • The slope is calculated as the ratio of change in y to the change in x (\( \Delta y / \Delta x \)).
In our equation \( y = 2x - 1 \), the slope is 2, meaning for every unit increase in x, y increases by 2 units. Understanding slope helps in predicting how the line behaves as you extend it across the coordinate plane.

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Most popular questions from this chapter

Graph each equation by constructing a table of values and then plotting the points. SEE EXAMPLE 5. (OBJECTIVE 4) $$x+y=-2$$

Graph each equation. SEE EXAMPLE 6. (OBJECTIVE 5 ) $$y=4$$

Express each direct variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 1. (OBJECTIVE 1) \(A\) varies directly with \(z\). If \(A=30\) when \(z=5,\) find \(A\) when \(z=9\).

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(x\) varies directly with \(c^{2}\) and inversely with \(f .\) If \(x=16\) when \(c=8\) and \(f=12,\) find \(x\) when \(c=10\) and \(f=5\).

Explain the process used to find the \(x\)- and \(y\)-intercepts of the graph of a line.
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