Problem 12 - Elementary Algebra

Question

Graph the equations. $$ y=-2 x+1 $$

Step-by-Step Solution

Answer

Answer: The slope of the equation y = -2x + 1 is -2, and the y-intercept is 1. To graph the equation, plot the y-intercept (0,1) on the coordinate plane, then use the slope (-2) to move 2 units down and 1 unit to the right from the y-intercept to find another point on the line (1,-1). Connect these points with a straight line, extending it in both directions, and label the line with the equation y = -2x + 1.

Step 1: Identify the slope and y-intercept

The given equation is in slope-intercept form: $$ y = -2x + 1 $$ In this form, the coefficient of x (m) represents the slope and the constant term (b) represents the y-intercept. In our case, the slope m=-2 and the y-intercept b=1.

Step 2: Plot the y-intercept

The y-intercept is the point on the graph where the line intersects the y-axis. The value of the y-intercept is given by b, which is 1 in this case. Therefore, the y-intercept is the point (0,1). To begin graphing the line, plot this point on your coordinate plane.

Step 3: Use the slope to plot another point

The slope of the line is -2 and represents the ratio of the change in y to the change in x (rise over run). In this case, the slope is -2/1. This means that from any point on the line, we can move 2 units down and 1 unit to the right to reach another point on the line. Starting at the y-intercept point (0,1), move 2 units down and 1 unit to the right. This will bring you to the point (1,-1). Plot this point on your coordinate plane.

Step 4: Draw the line and label it

Now that you have at least two points on the line (the y-intercept and the point found using the slope), connect the two points with a straight line using a ruler, extending the line in both directions. Label the line with the equation y = -2x + 1 to show that it represents the graph of that equation. In conclusion, the graph of the equation y = -2x + 1 is a straight line with a slope of -2 and a y-intercept of 1.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write linear equations so that they are easy to graph. This form is given by the equation \( y = mx + b \). In this equation, \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is where the line crosses the y-axis. Understanding this form is key:

• The slope \( m \) tells you how steep the line is.
• The y-intercept \( b \) tells you the specific point where the line touches the y-axis.

To graph a line using the slope-intercept form, you first plot the y-intercept on the y-axis. Then, using the slope, you determine the direction and steepness of the line to locate additional points.

Graphing Linear Equations

Graphing linear equations involves a few straightforward steps that help visualize the relationship between variables. Once you have the equation in slope-intercept form, plotting becomes easier. Follow these simple steps to graph:

• Locate the y-intercept on the graph and mark the point where the line will start.
• Using the slope, which is the rise over run (change in y over change in x), find the second point by moving from the y-intercept.
• Connect these points with a straight line extending it in both directions.

Each step builds on the last, allowing for accurate graphing of linear equations. Seeing the line on the graph helps in understanding how changes in the equation affect the graph’s shape.

Slope and Y-Intercept

The slope and y-intercept are two crucial components of linear equations. The slope, \( m \), indicates the inclination of the line

• If \( m \) is positive, the line rises as you move from left to right.
• If \( m \) is negative, the line falls.

The y-intercept, \( b \), is the value of \( y \) when \( x = 0 \). It is the starting point on the graph from which other points are calculated using the slope. In our example, the line has a slope of -2 and a y-intercept of 1, meaning it slopes downward and crosses the y-axis at (0,1). Knowing these two values makes it far easier to graph the equation quickly and accurately.

Coordinate Plane

The coordinate plane is the foundation for graphing equations. It consists of two number lines: the horizontal x-axis and the vertical y-axis, intersecting at a point called the origin, labeled \( (0,0) \). When graphing:

• Positive x-values are to the right of the origin, negative to the left.
• Positive y-values are above the origin, and negative are below.

Every point on the plane can be represented by a pair of numerical coordinates \( (x, y) \), which specify a location by determining how far along the x-axis and y-axis the point is located. Understanding the layout of the coordinate plane is vital for correctly plotting lines and interpreting the graphs of equations. It offers a visual way to understand mathematical concepts, making it easier to draw and interpret graphs.