Problem 2 - College Algebra

Question

Describe the process for fi ding the \(x\) -intercept and the \(y\) -intercept of a graph algebraically.

Step-by-Step Solution

Answer

Set y=0 for x-intercept; set x=0 for y-intercept.

Step 1: Understanding Intercepts

The x-intercept is the point where the graph of an equation crosses the x-axis. At this point, the y-coordinate is always zero. Conversely, the y-intercept is the point where the graph crosses the y-axis, where the x-coordinate is always zero.

Step 2: Finding the x-intercept

To find the x-intercept of a graph algebraically, set the y-value in the equation equal to zero and solve for x. For example, if the equation is given by \(y = 2x + 3\), set \(y = 0\) to get \(0 = 2x + 3\). Then solve for \(x\): \(2x = -3\) resulting in \(x = -\frac{3}{2}\). The x-intercept is \((-\frac{3}{2}, 0)\).

Step 3: Finding the y-intercept

To find the y-intercept, set the x-value in the equation to zero and solve for y. Using the same equation \(y = 2x + 3\), substitute \(x = 0\) to solve: \(y = 2(0) + 3\), thus \(y = 3\). The y-intercept is \((0, 3)\).

Key Concepts

Understanding the X-Intercept

The x-intercept is an important concept when graphing equations. It is the point where a graph crosses the x-axis. At this spot, the value of y is always zero because the graph hasn't moved up or down vertically. So, when you calculate x-intercept, you're finding the x-value when y is zero. Here's how you do it:

• Replace the y in your equation with zero.
• Solve for x. This will often require you to rearrange the equation and solve like a typical algebra problem.

Consider a simple linear equation, like the one given: \( y = 2x + 3 \). By setting y to zero, the equation becomes \( 0 = 2x + 3 \). Solving for x, we rearrange to get \( 2x = -3 \), meaning \( x = -\frac{3}{2} \). Thus, the x-intercept here would be at the point \((-\frac{3}{2}, 0)\). Being able to determine x-intercepts helps in showing where visual graphs cross the x-axis, which is essential for understanding the behavior of graphs.

Understanding the Y-Intercept

The y-intercept is where the graph crosses the y-axis. At this point, the x-value is always zero because the graph hasn't moved side-to-side horizontally along the x-axis. Finding the y-intercept algebraically is straightforward and involves setting x to zero:

• Substitute zero in place of x in the equation.
• Solve for y, if necessary.

For the equation \( y = 2x + 3 \), we replace x with zero to find the y-intercept: \( y = 2(0) + 3 \). This simplifies neatly to \( y = 3 \), showing that the y-intercept is at the point \((0, 3)\). Understanding y-intercepts is crucial when graphing lines or curves, as they indicate where the graph will cross the y-axis.

Graphing Algebraically

Graphing algebraically involves using calculations to understand the behavior of equations on a graph. By algebraically determining intercepts, the task of drawing graphs becomes easier and more accurate. The intercepts are typically the first key features you'll plot on a graph. Here's how you utilize your intercepts in graphing:

• Determine x-intercepts by setting y to equal zero and calculate for x.
• Find y-intercepts by setting x to zero and solving for y.
• Plot these intercept points on a coordinate plane.

Once the intercept points are plotted, they can serve as anchors, helping you further understand the slope and behavior of the graph. In linear equations, these two points define the line across the graph. Remember, intercepts are your best tool for verifying if a graph has been drawn correctly and understanding the relationships between algebraic equations and their graphical representations.