Problem 7 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. Graph the equation. $$ y=x+2 $$

Step-by-Step Solution

Answer

The x-intercept is (-2,0) and the y-intercept is (0,2).

Step 1: Finding the Y-intercept

Set x equal to zero and solve for y. In the equation \(y = x+ 2\), if \(x = 0\), then \(y = 0 + 2\), so \(y = 2\).

Step 2: Finding the X-intercept

Set y equal to zero and solve for x. In the equation \(y = x + 2\), if \(y = 0\), then \(0 = x + 2\). After Juggling the equation to solve for x, \(x = -2\).

Step 3: Graphing the Equation

Plot the y-intercept (0,2) and the x-intercept (-2,0) on a graph. Next, draw the line connecting these two points to represent the equation \(y= x+2\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept for any linear equation, you can set y equal to zero and solve for x.
For example, take the equation \( y = x + 2 \). By setting y to zero, you have:

• \( 0 = x + 2 \)
• Subtract 2 from both sides: \( x = -2 \)

This tells us that the x-intercept is \((-2, 0)\), which means the graph crosses the x-axis at this point.
By understanding the x-intercept, you can quickly identify one distinct point that will help you in graphing linear equations.

Exploring the Y-Intercept

The y-intercept is where the graph crosses the y-axis. In this instance, the x value is zero. To find a y-intercept, substitute zero for x in the equation and solve for y.
Using the equation \( y = x + 2 \):

• Replace x with zero: \( y = 0 + 2 \)
• Simplifying gives you \( y = 2 \)

So the y-intercept is \((0, 2)\), which is where the line crosses the y-axis.
Knowing how to find the y-intercept is essential. It provides another anchor point for sketching the graph of a linear equation. Additionally, the y-intercept often represents the starting point in real-life applications where x represents time or another independent variable.

The Art of Plotting Points

Plotting points on a graph involves marking specific points on the coordinate plane based on their coordinates. Each point is defined by an (x, y) pair, making it very tangible and visual.
To plot the graph of \( y = x + 2 \), you can start with the intercepts:

• Plot the y-intercept at \((0, 2)\), noting this is where the line crosses the y-axis.
• Plot the x-intercept at \((-2, 0)\), the point where the line crosses the x-axis.

To confirm accuracy, it's always a good idea to explore more points. For instance, substitute random x-values and solve for their corresponding y-values, then plot these points:
Consider \( x = 1 \):

• Substitute into the equation: \( y = 1 + 2 = 3 \)
• Plot the point \((1, 3)\)

Finally, draw a straight line through all plotted points to visualize the entirety of the linear equation. This approach not only aids understanding but also ensures the accuracy of your graph.