Problem 25 - Thinking Mathematically

Question

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=2 x+1\)

Step-by-Step Solution

Answer

The points obtained are \((-3, -5), (-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5), (3, 7)\). When plotted and connected, these points form a straight line which represents the equation \(y=2 x+1\) on the graph.

Step 1: Compute y-coordinates for given x-values

For each of the \(x\) values between \(-3\) and \(3\), substitute the \(x\) value into the equation to find the corresponding \(y\) value. This will provide you with a set of points that can be plotted on a coordinate plane.

Step 2: List of Points for Graphing

After computing the y-values, obtain the list of points as follows: \((-3, -5), (-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5), (3, 7)\)

Step 3: Plotting the Points on a Graph

Plot these points on a graph. The x-values run along the horizontal axis and the y-values along the vertical axis. The point (0,1) represents where the line crosses the y-axis (also known as the y-intercept), and the slope is 2 (which means for every step right we go on the x-axis, we go two steps up on the y-axis)

Step 4: Draw the Line

Once all the points are plotted, connect them to draw the line representing \(y=2 x+1\). This line is the graphic representation of the equation, showing all the possible solutions. The line should be straight, indicative of a linear equation.

Key Concepts

The Coordinate Plane

Imagine a flat surface on which you can pinpoint locations, much like marks on a map. The coordinate plane, also known as the Cartesian plane, is that map for mathematics. It's composed of two number lines that intersect at a right angle. The horizontal number line is called the x-axis, and the vertical one is called the y-axis. Their intersection point marks the origin, designated as (0,0).

Every point on this plane can be described by a pair of numbers, referred to as coordinates, which reveal its location in relation to the origin. The first number, or the x-coordinate, tells how far to move right (positive) or left (negative) from the origin. The second number, the y-coordinate, indicates how far to move up (positive) or down (negative). The coordinates are written as (x, y). This system allows us to precisely plot points, and consequently, graph equations.

Slope-Intercept Form

The slope-intercept form is invaluable for graphing linear equations quickly and is written as \(y = mx + b\). Here, \(m\) stands for the slope of the line, which tells us how steep the line is and in which direction it tilts. The \(b\) stands for the y-intercept, where the line crosses the y-axis.

The beauty of this form lies in its straightforward nature. From the equation \(y = 2x + 1\), for instance, we can see that the slope is 2, indicating that for each unit we move right along the x-axis, we should move up two units (because the slope is positive). Also, we have the y-intercept at 1, meaning the line will cross the y-axis at (0,1).

Plotting Points

To plot points, we begin with the prepared list of coordinates. Each coordinate dictates where to place a single point on the coordinate plane. Start at the origin, then follow the x-coordinate right or left and the y-coordinate up or down. Once you reach the destination, mark a point.

For instance, using our given equation \(y = 2x + 1\), one of the coordinates we've calculated is (-1, -1). Starting at the origin, we'd move 1 unit left (since the x-coordinate is -1), and then 1 unit down (as the y-coordinate is also -1), and plot the point there. Repeat the process for each coordinate until you have them all plotted.

Linear Equation

A linear equation is an equation that makes a straight line when graphed on the coordinate plane. The standard form is \(Ax + By = C\), but as we've seen, the slope-intercept form is more useful for graphing. A line is described by its slope and y-intercept, properties comprehensively captured in the slope-intercept form.

The equation \(y = 2x + 1\) is linear because it graphs as a straight line, the hallmark characteristic of linear equations. Since every x-value has only one corresponding y-value, the graph of these kinds of equations will always result in a straight line without curves or bends.