Problem 28 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The graph of the equation \(y=-\frac{1}{2} x+2\) is a straight line that slopes downwards, passing through the points \((-3, 3.5)\), \((-2, 3)\), \((-1, 2.5)\), \((0, 2)\), \((1, 1.5)\), \((2, 1)\), \((3, 0.5)\).

Step 1: Determine the Slope

The slope of the line, \(m = -\frac{1}{2}\), means that for every step to the right on the x-axis, the line goes down by \(-\frac{1}{2}\) on the y-axis.

Step 2: Determine the Y-intercept

The equation gives us the y-intercept directly as 2. This means that the line will pass through the point (0,2) on the graph.

Step 3: Plot Integers for X

You need to substitute integers from -3 to 3 into the equation and calculate the corresponding y-values. You will get points \((-3, 3.5)\), \((-2, 3)\), \((-1, 2.5)\), \((0, 2)\), \((1, 1.5)\), \((2, 1)\), \((3, 0.5)\).

Step 4: Plot the Line

Plot the points found in the previous step on a graph. Draw a line that passes through these points to represent the equation.

Key Concepts

Slope of a Line

Understanding the slope of a line is vital when graphing linear equations. The slope, often represented by the letter m, measures how steep a line is. It is calculated as the rise over run, which means the change in y (vertical) divided by the change in x (horizontal).

In the given exercise, the slope of the line is \( m = -\frac{1}{2} \). This indicates that for every one unit you move to the right on the x-axis (run), the value on the y-axis decreases by half a unit (rise), signifying a downward slope. Think about it as a small hill that tilts downwards. An upward slope would mean an increase in y-values as x increases.

In practice, you visually assess how steep your line should be when plotting it on a graph. Imagine walking on the graph from one plotted point to the next: For a negative slope like ours, you'd be going downhill for each step to the right!

Y-intercept

The y-intercept is another fundamental concept in graphing linear equations. It's where the line crosses the y-axis. The y-intercept is easy to spot in most linear equations, as it is the y value when x equals zero. In other words, it's the starting point of the line on the graph when reading from left to right.

In our exercise, the y-intercept is explicitly stated in the equation: \( y = -\frac{1}{2} x + 2 \), where the number 2 is the y-intercept. Hence, the point (0,2) is where the line will touch the y-axis. This point is your anchor; when you begin plotting, you start here and then use the slope to determine the direction and steepness as you plot additional points.

Plotting Points on a Graph

To accurately represent a linear equation, we must plot points on a graph. Plotting points is like creating a map of values that your line will follow. You'll start with the y-intercept and then use the slope to find other points along the line.

With our exercise equation, \( y=-\frac{1}{2} x+2 \), you chose integers for x between -3 and 3. By substituting these values into the equation, you found corresponding y-values, giving you coordinates to plot. For example, inserting x = 1 gives us \( y = -\frac{1}{2} (1) + 2 = 1.5 \), so you plot the point (1, 1.5).

Repeat this step for all selected x-values to get a series of points. When these points are plotted on a graph and connected, they form the visual representation of the equation - in this case, a straight line.

Linear Equation Representation

A linear equation representation combines all previous concepts into a coherent image on a graph. Linear equations are typically written in the form \( y = mx + b \), where m is the slope and b is the y-intercept. This standard form simplifies the process of graphing by giving you the initial point (the y-intercept) and the direction and steepness of the line (the slope).

Once you have the points plotted on a graph, it's time to draw the line. It should pass through all the points, extending beyond them in both directions, to represent all possible solutions for x and y. Your line is the visual representation of every solution to the equation. It's where the algebraic expression of the equation takes physical shape, showing the direct relationship between x and y.