Problem 2
Question
For each equation make a table of point pairs, taking integer values of \(x\) from -3 to 3, plot these points, and connect them with a smooth curve. $$y=2 x-2$$
Step-by-Step Solution
Verified Answer
A table of values with x ranging from -3 to 3 and corresponding y-values using the equation y=2x-2, plotted on a graph and connected with a straight line.
1Step 1: Create a Table of Values
Start by creating a table with two columns, one for integer values of x and one for the corresponding y-values. Choose integer values of x ranging from -3 to 3. For each x-value, calculate the y-value using the equation y = 2x - 2 and fill in the table.
2Step 2: Calculate y-values
Use the equation y = 2x - 2 to find the y-values for each chosen x-value. For example, when x = -3, y = 2(-3) - 2 = -8. Continue this process for all x-values from -3 to 3.
3Step 3: Plot the Points
On a graph with an x and y-axis, plot each point pair (x, y) from the table. Ensure that each point is plotted accurately according to the scale of your graph.
4Step 4: Connect the Points
After plotting all the points on the graph, use a ruler to connect them with a smooth line. Since the equation is linear, the points should form a straight line.
Key Concepts
Table of ValuesLinear FunctionsGraphing Techniques
Table of Values
Understanding a linear equation can often start with creating a table of values. It's essentially like setting up a roadmap for what you're about to plot on a graph. Think of this table as a translator between a mathematical equation and a visual representation. To do this, you first select a range of x-values – in our exercise, we choose integers from -3 to 3.
Then, for each x-value, we find the corresponding y-value using the given equation, which in this case is the linear equation \(y = 2x - 2\). This process forms 'point pairs', or coordinates, which will later take their place on the graph. You start by simply substituting the x-value into the equation and calculating to find y. For instance, when x=1, y becomes 2*1-2, which equals 0. Hence, one of the points you will use is (1, 0). Repeating this for each x-value gives us a complete table, ready to act as a blueprint for plotting the graph.
Then, for each x-value, we find the corresponding y-value using the given equation, which in this case is the linear equation \(y = 2x - 2\). This process forms 'point pairs', or coordinates, which will later take their place on the graph. You start by simply substituting the x-value into the equation and calculating to find y. For instance, when x=1, y becomes 2*1-2, which equals 0. Hence, one of the points you will use is (1, 0). Repeating this for each x-value gives us a complete table, ready to act as a blueprint for plotting the graph.
Linear Functions
When we talk about linear functions, we refer to one of the most fundamental algebraic expressions encountered in math. These functions form a straight line when graphed and are represented by the general form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In our scenario, the function \(y = 2x - 2\) is a linear function with a slope of 2—meaning for every unit increase in x, y increases by 2 units—and a y-intercept where the line crosses the y-axis, which is at -2. What's crucial here is to note that the value of 'm' determines how 'steep' the line is. In practice, the linearity ensures that the relationship between x and y is consistent, with the y-values changing at a constant rate as x changes. As this function is first-degree, meaning the highest power of x is 1, the graph will indeed be a straight line, not a curve.
In our scenario, the function \(y = 2x - 2\) is a linear function with a slope of 2—meaning for every unit increase in x, y increases by 2 units—and a y-intercept where the line crosses the y-axis, which is at -2. What's crucial here is to note that the value of 'm' determines how 'steep' the line is. In practice, the linearity ensures that the relationship between x and y is consistent, with the y-values changing at a constant rate as x changes. As this function is first-degree, meaning the highest power of x is 1, the graph will indeed be a straight line, not a curve.
Graphing Techniques
Graphing is the art of bringing numbers to life on paper - think of it as plotting the constellations of the math world. To transform the table of values into a visual display, you start on graph paper, or a digital equivalent, with two perpendicular lines known as the x and y axes. Each point pair, or coordinate, you've found in your table, represents a specific point on this grid.
For example, the point (1, 0) means you move one step along the x-axis and since y is 0, you stay at the origin of the y-axis. Once you plot all the points, you use graphing techniques such as ensuring the use of a straightedge to draw the line through these points. Because we're dealing with linear functions, you'll find your points line up neatly, allowing you to draw a single straight line extending through them, ideally continuing beyond the plotted points to cover all possible x-values. This visualization is not only confirming the linear nature of our function but also serves as a practical tool for predicting values and understanding the relationship between variables in the function.
For example, the point (1, 0) means you move one step along the x-axis and since y is 0, you stay at the origin of the y-axis. Once you plot all the points, you use graphing techniques such as ensuring the use of a straightedge to draw the line through these points. Because we're dealing with linear functions, you'll find your points line up neatly, allowing you to draw a single straight line extending through them, ideally continuing beyond the plotted points to cover all possible x-values. This visualization is not only confirming the linear nature of our function but also serves as a practical tool for predicting values and understanding the relationship between variables in the function.
Other exercises in this chapter
Problem 2
Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=4-2 x^{2}$$
View solution Problem 2
If \(h\) and \(k\) are positive quantities, in which quadrants would the following points lie? $$(h, k)$$
View solution Problem 3
Find the slope of each straight line. Rise \(=-4.25,\) run \(=5.33\)
View solution Problem 3
Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=3 x-2$$
View solution