Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=1-2 x$$

Short answer

The graph of the function \(y = 1 - 2x\) is a straight line with a y-intercept of (0, 1) and a slope of -2. Plot the y-intercept and use the slope to find another point, then draw the line through these points.

Step-by-step solution

Understand the Equation

The given equation is a linear function in the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. In this case, the slope \(m\) is -2, and the y-intercept \(b\) is 1.

Determine the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In the given function, \(y = 1 - 2x\), the y-intercept is (0, 1). Plot this point on the graph.

Use the Slope to Find Another Point

Since the slope is -2, it means for every one unit we move horizontally to the right, we move two units vertically down. Starting from the y-intercept (0, 1), move 1 unit to the right to x = 1 and 2 units down to y = -1. This gives us another point on the line: (1, -1). Plot this point as well.

Draw the Line

Draw a straight line through the two points, extending it across the initial viewing window from -5 to 5 for both the x and y axes. If the line does not fit well in the window, adjust the scale to better display the line.

Key concepts

Linear Equations

A linear equation is a fundamental concept in algebra, representing a straight line when graphed on a coordinate plane. These equations are typically written in the standard form ( y = mx + b ), where m is the slope of the line and b is the y-intercept — the point where the line crosses the y-axis. The beauty of linear equations lies in their simplicity and the straightforward method for graphing them.

For our exercise, the linear equation is ( y = 1 - 2x ). By analyzing this, we establish that the equation follows the standard form, indicating a slope (m) of -2 and a y-intercept (b) of 1. Understanding this foundational structure allows students to graph the function easily, providing a visual representation of the equation.

Y-Intercept

The y-intercept of a linear equation is a crucial point where the line passes through the y-axis, and it's always at an x-value of 0. This point is valuable for graphing as it provides a starting position from which you can use the slope to find other points on the line. In the equation ( y = 1 - 2x ), the y-intercept is ( b = 1 ). This means that the point (0, 1) is where our line will intersect the y-axis.

Plotting the y-intercept on a graph is the first step in visualizing the function. The clarity of locating the y-intercept lies in setting x equal to zero and solving for y, which always gives the exact spot on the y-axis where the line will cross.

Slope

The slope of a linear function measures the steepness and direction of a line. It is defined as the ratio of the change in vertical height (rise) to the change in horizontal distance (run) between any two points on the line. The slope can be positive, negative, or zero, telling us if the line is inclining, declining, or horizontal.

In our example, with the equation ( y = 1 - 2x ), the slope is -2. This negative slope indicates that the line descends as we move from left to right across the graph. For every 1 unit increase in x, the value of y decreases by 2 units, showcasing a downhill pattern. Understanding slope is significant—it not only determines the angle of the line but also helps us to plot additional points after the y-intercept for a more accurate graph.

Plotting Points

Plotting points is a methodical approach to create a graph representing a linear equation. After finding the y-intercept, you use the slope to determine the next points. You continue this process systematically: from each new point, apply the slope by moving right or left for x, and up or down for y, depending on the sign and magnitude of the slope.

In the current exercise, after plotting the y-intercept (0, 1), we apply the slope -2. From the y-intercept, we move 1 unit to the right (the run) and 2 units down (the fall) since the slope is negative. The next point would be at (1, -1). By connecting these points with a straight line and extending this line across the graph, we visualize the linear equation. Repeating this slope movement as needed, we can plot as many points as necessary to ensure our line is accurate.