Problem 76 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graph of each equation. $$ f(x)=-2 x-1 $$

Step-by-Step Solution

Answer

Graph is a straight line with y-intercept at (0, -1) and slope of -2.

Step 1: Identify the Type of Function

The given equation is in the form of a linear function:\[ f(x) = -2x - 1 \]A linear function graphs to a straight line.

Step 2: Find the Slope and Y-Intercept

Rewrite the equation in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.In this case, the slope (\( m \)) is -2 and the y-intercept (\( b \)) is -1.

Step 3: Plot the Y-Intercept

The y-intercept is the point where the graph of the equation crosses the y-axis.For this equation, the y-intercept is at \( (0, -1) \). Plot this point on the graph.

Step 4: Use the Slope to Find Another Point

The slope of -2 means that for every change of 1 in \( x \), \( y \) changes by -2.Starting from the y-intercept (0, -1), move 1 unit to the right (increase \( x \) by 1) to \( x = 1 \), and then move 2 units down (since the slope is negative) to \( y = -3 \). Plot the point \( (1, -3) \).

Step 5: Draw the Line

Draw a straight line passing through the points \( (0, -1) \) and \( (1, -3) \). This line represents the graph of the function \( f(x) = -2x - 1 \). Ensure that the line extends across the graph to illustrate the linearity.

Key Concepts

Slope-Intercept Form

In mathematics, understanding the slope-intercept form of a linear equation is key to graphing linear functions. The slope-intercept form is written as: \( y = mx + b \). This form is powerful because it directly tells us two critical aspects of the line:

• The slope \( (m) \)
• The y-intercept \( (b) \)

Knowing these two elements allows one to quickly graph an equation without needing to calculate any additional points. In our original exercise, the equation \( f(x) = -2x - 1 \) is already in this form. This tells us that the slope \( m \) is -2, and the y-intercept \( b \) is -1. Grasping this form simplifies the graphing process significantly.

Graphing Equations

Graphing linear equations involves plotting them on a coordinate plane to visualize the relationship between variables. First, identify the slope and y-intercept from the equation in slope-intercept form. With the y-intercept found, plot this on the y-axis. Then use the slope to determine the direction and steepness of the line. For example, a slope of -2 indicates that for every 1 unit increase to the right on the x-axis, the line will decrease by 2 units on the y-axis. This process allows us to draw a straight line, as the name 'linear' suggests, giving a straight passage through all plotted points. Once you have two points, use a ruler to draw a line extending through them, making sure to maintain consistent slope throughout.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the equation \( y = mx + b \), \( b \) represents the y-intercept. To find this point, set \( x \) to zero and solve for \( y \). In our case, this is accomplished simply by looking at \( b \) in \( f(x) = -2x - 1 \), indicating the point \( (0, -1) \) is the y-intercept. Plot this point on the graph to serve as a starting point for drawing the line.Once placed, the y-intercept offers a crucial reference for drawing and understanding the entire graph of the line.

Slope of a Line

The slope of a line defines its steepness and direction. It is represented by the symbol \( m \) in the equation \( y = mx + b \).The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In our exercise, the slope is -2, which translates to a fall of 2 units for each unit the line moves to the right. Calculating the slope involves determining the amount \( y \) changes for a given change in \( x \), commonly described as "rise over run." Understanding the slope helps you plot the second point after plotting the y-intercept. From the y-intercept \( (0, -1) \), you move 1 unit to the right on the x-axis, and 2 units down on the y-axis, plotting the point \( (1, -3) \).With these two points, you can draw the entire line, capturing the basic essence of the linear function's behavior.