Problem 20 - Beginning and Intermediate Algebra

Question

Graph each function. $$g(x)=-x+3$$

Step-by-Step Solution

Answer

To graph the function \(g(x) = -x + 3\), first identify the slope (m) as -1 and the y-intercept (b) as 3. Plot the y-intercept (0, 3) on the y-axis. Using the slope, find another point on the line by moving one unit right and one unit down from the y-intercept to get the point (1, 2). Plot this point and connect both points with a straight line, representing the graph of the function \(g(x) = -x + 3\).

Step 1: Identify the slope and y-intercept

To graph any linear function in the form \(y = mx + b\), we need to identify the slope (m) and the y-intercept (b). In this case, our function is \(g(x) = -x + 3\), so the slope (m) is -1 and the y-intercept (b) is 3.

Step 2: Plot the y-intercept

The y-intercept is the point where the line crosses the y-axis. In our case, the y-intercept is 3, which corresponds to the point (0, 3). Plot this point on the y-axis on your graph.

Step 3: Use the slope to find another point

We can use the slope to find another point on the line. The slope is -1, which means that for every 1 unit we move in the x-direction, the line will decrease by 1 unit in the y-direction. From the y-intercept (0, 3), we can go one unit to the right and one unit down to get the point (1, 2). Plot this point on your graph.

Step 4: Draw the line

Now that we have two points, (0, 3) and (1, 2), connect them with a straight line. This is the graph of the function \(g(x) = -x + 3\). Keep in mind that the line extends infinitely in both directions.

Key Concepts

Slope and Y-intercept

The slope and y-intercept are essential concepts when graphing linear functions. The slope, often represented as \( m \), tells us how steep a line is and the direction it t rends. In our exercise, the slope is \(-1\). This means the line goes downward from left to right, indicating a negative slope. The steeper the slope, the greater the change in the y-values for a given change in x-values. The y-intercept is the point where the line crosses the y-axis. It's represented by \( b \) in the linear equation \( y = mx + b \). In this exercise, the y-intercept is 3. This means that when the x-value is 0, the y-value is 3. Plotting the y-intercept is the starting point for any graph of a linear function.

Plotting Points

Plotting points involves placing points on a graph based on their coordinates. When you know the slope and y-intercept of a linear function, you can quickly determine two points, which is all you need to draw the line. Start by plotting the y-intercept. In our example, the first point is (0, 3).To find the second point, use the slope. From (0, 3), move one unit to the right and one unit down (because the slope is \(-1\)) to find the second point, (1, 2). These points give you an accurate guide for drawing your line.

Linear Equations

Linear equations describe lines on a graph. They have the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. They are called 'linear' because their graph is a straight line.Think of a linear equation as a precise set of instructions for drawing a line. Just by knowing the slope and y-intercept, you can recreate the line on a coordinate plane. The beauty of linear equations lies in their simplicity and predictability, making them an essential part of understanding algebra.

Coordinate Plane

Understanding the coordinate plane is crucial when graphing functions. It's like a map for plotting points formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Every point on this plane is defined by an (x, y) coordinate. The origin, where these axes intersect, is at (0, 0). In our task, plotting (0, 3) and (1, 2) involves locating these coordinates in relation to the origin. The coordinate plane is vital as it allows us to visualize relationships between variables, which is a core skill in graphing linear equations.