Problem 27 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The graph of the equation \(y=-\frac{1}{2} x\) is a straight line that passes through the points (-3,1.5), (-2,1), (-1, 0.5), (0,0), (1,-0.5), (2,-1), and (3,-1.5).

Step 1: Substitute value of \(x\) in the equation

First, plug in the values of \(x\) between -3 and 3 into the equation \(y=-\frac{1}{2} x\) one by one and solve for \(y\). You get the following pairs of coordinates: \[\begin{align*}(-3,1.5),\(-2,1),\(-1, 0.5),\(0,0),\(1,-0.5),\(2,-1),\(3,-1.5)\end{align*}\]

Step 2: Plot the coordinates

Next, plot the points that you obtained in the previous step on the coordinate plane. Start with the y-intercept which is (0,0). Then plot the rest of the points.

Step 3: Draw the line

Once the points are plotted, draw a straight line that passes through these points. This line is the graph of the linear equation \(y=-\frac{1}{2} x\).

Key Concepts

Plotting Coordinates

Understanding how to plot coordinates is vital for graphing linear equations. To begin, consider each pair of numbers as directions for reaching a specific spot on a map—this 'map' is known as the coordinate plane. Starting at the origin (0,0), you move to the right for a positive first number, or to the left for a negative. Then, go up for a positive second number, or down for a negative. This process helps visualize the relationship between numbers in an equation.

Imagine you’re plotting the point (-2,1). You would move two spaces to the left of the origin and then one space up. Repeating this process for all coordinate pairs derived from the equation provides us with a series of points which, when connected, will display the shape of the equation on the graph.

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations produce straight lines when graphed on a coordinate plane. They have the general form of \(y = mx + b\), where \(m\) represents the slope and \(b\) denotes the y-intercept—the point where the line crosses the y-axis.

In the example \(y = -\frac{1}{2} x\), the equation tells us that for every unit change in \(x\), \(y\) will decrease by half a unit. This information can be used to generate the points to be plotted. It is this consistent change that gives a straight line graph, showcasing the uniform rate of change typical of linear relationships.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can map numbers and, as a result, functions like our linear equations. It's made up of two perpendicular axes: the horizontal is called the x-axis, and the vertical is the y-axis.

The point where these lines intersect is the origin. Every point on the plane is determined by an \((x, y)\) coordinate which shows its horizontal and vertical distances from the origin. To effectively use the coordinate plane, remember that it's divided into four quadrants, and understanding these quadrants is crucial when plotting coordinates that result from inserting different values for \(x\) into the linear equation.

Slope

When graphing a line, the slope is a measure of how steep the line is. It is calculated as 'rise over run', representing the change in the y values (rise) over the change in the x values (run). Positive slope slants upward to the right, while a negative slope slants downward to the right.

Understanding Slope in Linear Equations

In the equation \(y = -\frac{1}{2} x\), the slope is \(-\frac{1}{2}\). This means for every 2 units you move to the right (run), you move 1 unit down (rise). Knowing the slope helps us predict and understand the direction and steepness of the line without plotting all points, and it ensures that connecting plotted points will result in a straight line, accurately representing the linear equation on the graph.