Problem 3 - Technical Mathematics with Calculus

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. $$y=3-2 x$$

Step-by-Step Solution

Answer

The points on the graph are (-3, 9), (-2, 7), (-1, 5), (0, 3), (1, 1), (2, -1), and (3, -3). When connected, they form a straight line with a slope of -2 and a y-intercept of 3.

Step 1: Create a Table of Values

To plot the equation, start by creating a table of values by substituting integer values of x from -3 to 3 into the equation y = 3 - 2x, and solve for y to find the corresponding y-values.

Step 2: Calculate y for Each x Value

Substitute the x values into the equation to get the y values: For x = -3, y = 3 - 2(-3) = 9; For x = -2, y = 3 - 2(-2) = 7; For x = -1, y = 3 - 2(-1) = 5; For x = 0, y = 3 - 2(0) = 3; For x = 1, y = 3 - 2(1) = 1; For x = 2, y = 3 - 2(2) = -1; For x = 3, y = 3 - 2(3) = -3.

Step 3: Plot the Points

Take each (x, y) pair from the table and plot it on a coordinate grid. The points are (-3, 9), (-2, 7), (-1, 5), (0, 3), (1, 1), (2, -1), and (3, -3).

Step 4: Connect the Points

After plotting all the points, draw a smooth curve through them to represent the equation y = 3 - 2x. Since this is a straight line, use a ruler to make sure the line is straight.

Key Concepts

Table of Values

When plotting linear equations, a table of values is an essential starting point. It's essentially a chart where you list pairs of numbers that correspond to the 'x' and 'y' values for the points on a graph. To create one, you choose a range of values for 'x' and use the equation to find the corresponding 'y' values.

Having a table of values makes it easier to visualize how the 'y' value changes as 'x' increases or decreases. For our example equation, \(y = 3 - 2x\), we select integer values of 'x' from -3 to 3, and for each 'x', we calculate the 'y'. Here's a simplified approach to what this process looks like:

• Pick a value for 'x'.
• Substitute it into the equation.
• Solve for 'y'.
• Record the pair \((x, y)\) in the table.

Once you have all the pairs listed, you’re ready to move on to plotting them on a grid.

Coordinate Grid

A coordinate grid is a two-dimensional plane made up of a horizontal line (x-axis) and a vertical line (y-axis). These axes intersect at a point called the origin, marked as (0,0). Our task is to use the grid as a map for plotting the points from our table of values.

Each point \((x, y)\) corresponds to a position on this grid. For instance, the point (-3, 9) means you move 3 units to the left (because it's negative) on the x-axis and then move up 9 units on the y-axis. Plotting all the points from the table gives us a series of dots that will eventually be connected to show the graph of the equation.

Linear Equation Graphing

Graphing a linear equation like \(y = 3 - 2x\) results in a straight line. The process is straightforward after you have a table of values and a coordinate grid ready to go. Here's what you need to do:

• Plot the points from your table on the grid. Each \((x, y)\) pair gives the exact location of a point on the graph.
• Once all points are plotted, you'll notice they align in a straight path.
• Use a ruler to draw a line through the points, extending it in both directions beyond the plotted points.

This line is the visual representation of all the solutions to the equation and it clearly shows how 'y' is affected by 'x'.

Substituting Values

The method of substituting values is crucial for filling out the table of values mentioned earlier. Every time we substitute an 'x' into the equation, we find the corresponding 'y'. It’s like solving a mini puzzle: if 'x' is this, what is 'y'?

This simple act of replacing 'x' with a number and calculating 'y' is the crux of plotting linear equations. It requires some basic arithmetic. For example, when 'x' is 2, we substitute it into the equation to get \(y = 3 - 2(2)\), simplifying to \(y = -1\). We repeat this process for each value of 'x' in our chosen range, building up our complete table of values to graph.