Problem 41 - College Algebra

Question

Give the slope and y-intercept of each line whose equation is given. Then graph the line. $$y=-2 x+1$$

Step-by-Step Solution

Answer

The slope of the line is -2 and the y-intercept is 1. The line passes through the points (0,1), (1,-1), (2,-3), etc.

Step 1: Identify the Slope

The slope (m) is the coefficient of x in the equation. In the equation \(y = -2x + 1\), the slope is -2.

Step 2: Identify the Y-Intercept

The y-intercept (b) is the constant term in the equation. In the equation \(y = -2x + 1\), the y-intercept is 1.

Step 3: Draw the Graph

Plot the y-intercept point on the y-axis, which is (0,1). The slope -2 implies that for every step to the right (increase x by 1), you go two steps down (decrease y by 2). This can be continued to get more points like (1,-1), (2,-3), etc. Plot these points and draw a straight line passing through them.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial for anyone dealing with linear equations. The slope indicates how steep a line is. It’s determined by the ratio of the 'rise' to the 'run' between any two points on the line. In mathematical terms, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a graph, the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For the simple equation \( y = -2x + 1 \), the slope is -2. This means for every unit we move right along the x-axis, we move 2 units down in the y-axis direction, as the slope is negative. This concept is fundamental not only in plotting the graph but also in understanding the line's direction. A positive slope ascends from left to right, while a negative slope descends in the same direction.

Rate of Change

Think of the slope as the line's rate of change. It reflects how much \( y \) changes in response to a change in \( x \). In real-world terms, if you were hiking, the slope would represent the incline's steepness.

Y-Intercept

The y-intercept is another integral component of graphing linear equations. It is the point where the line crosses the y-axis. Technically, this is where \( x=0 \). The y-intercept provides a convenient starting place for plotting the line on a graph, as you always know one point the line will pass through.

From the equation \( y = -2x + 1 \), the y-intercept is 1. This is expressed as the point \( (0, 1) \). It's a key anchor for drawing the entire line because once you have the y-intercept, you can use the slope to find other points.

Importance in Equations

The y-intercept is particularly useful in solving various real-life problems where the initial value of a dataset is required. It often represents the starting point, or the initial condition of a scenario being modeled by the equation.

Plotting Points on a Graph

Plotting points is the action of marking dots on a graph at coordinates that match the variables in an equation. It translates the abstract equation into a visual representation that is much easier to interpret and use. When plotting points for a linear equation like \( y = -2x + 1 \), we begin with the y-intercept, which is (0, 1) in this case.

Then, we use the slope to find other points. Since our slope is -2 for the equation \( y = -2x + 1 \), starting from the y-intercept, one unit move right (increase in \( x \) value) will result in a two-unit downward move (decrease in \( y \) value). Hence, you can plot (1, -1), (2, -3), and so forth. By connecting these points with a straight line, the graph of the equation is completed.

Practical Application

Practicing plotting points helps in understanding functions and interpreting data on charts, necessary skills in many fields like economics, physics, and social sciences.