Problem 39 - Algebra 1

Question

Graph the line that has the given intercepts. $$ \begin{array}{l} x \text { -intercept: }-12 \\ y \text { -intercept: }-8 \end{array} $$

Step-by-Step Solution

Answer

Draw a line passing through the x-intercept point (-12,0) and the y-intercept point (0,-8). This is the graphical solution.

Step 1: Understanding X and Y intercepts

An intercept is the point where the line crosses an axis. The x-intercept is the value of x when y = 0. Here, the x-intercept is given as -12, i.e., this is the point (-12,0) where the line crosses the x-axis. Similarly, the y-intercept is the value of y when x = 0. Here, the y-intercept is given as -8 which is the point (0,-8) where the line crosses the y-axis.

Step 2: Plotting the intercepts

Plot these two intercept points (-12,0) and (0,-8) on the coordinate plane.

Step 3: Drawing the line

Draw a line that passes through both intercept points. This represents the equation that includes both x-intercept and y-intercept.

Key Concepts

Understanding the X-Intercept

The x-intercept is a crucial part of graphing linear equations. To find it, you determine where a line crosses the x-axis on a graph. This crossing point is significant because at this location, the value of \( y \) is zero. For instance, in our exercise, the x-intercept is given as -12. This means that the point (-12, 0) is where the line meets the x-axis.
Understanding the x-intercept helps us place a 'fixed' point on the graph, which simplifies the process of drawing a line. By knowing this coordinate, you can better predict the behavior and path of the line on the coordinate plane. Keep in mind:

• It always has the form \((x,0)\)
• Helps in identifying the slope of the line when paired with another point
• Is essential for forming the equation of the line

Exploring the Y-Intercept

As with the x-intercept, the y-intercept is another vital component of linear equations. It indicates the point at which the line crosses the y-axis. At this point, the value of \( x \) is zero. In our specific exercise, the y-intercept is -8, which means the line meets the y-axis at the point (0, -8).
Recognizing the y-intercept allows you to understand the starting point of the line when graphed, assuming \( x \) starts from zero. This intercept is important for establishing both the direction and steepness of the line on the graph. Remember:

• The y-intercept is in the format \((0,y)\)
• This point can help derive the line's equation
• Along with the x-intercept, it provides two definitive points to draw the line

Navigating the Coordinate Plane

The coordinate plane is essentially a two-dimensional graph that helps visualize mathematical concepts. It's composed of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Lines and shapes are represented within this grid, making it easier to perform and understand calculations.
For our exercise, plotting points on this plane provides a visual reference to where the x and y intercepts are located. A couple of key features include:

• The origin is where both axes meet at point (0,0)
• Each axis is used to plot number values, positive to the right and up, negative to the left and down
• Helps in understanding how to graph always by plotting intercepts as direct points

The coordinate plane thus becomes an essential tool in graphing, letting you navigate the relationship between different points and lines comprehensively. With a grasp of the plane's grid structure, one can effectively determine the path of lines, ensuring more accurate graphing of linear equations.