Problem 2 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Set y=0 to find the x-intercept, and set x=0 to find the y-intercept.

Step 1: Understand the Problem

To find the intercepts of a graph algebraically, you need to locate where the graph intersects the x-axis and y-axis. The x-intercept is the point where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.

Step 2: Find the x-intercept

To find the x-intercept, set the y-value to zero in the equation of the line, since at the x-intercept, the graph crosses the x-axis where y is zero. Solve the equation for x. For example, if the equation of the graph is \( y = 2x + 3 \), set \( y = 0 \) and solve: \[ 0 = 2x + 3 \] \[ 2x = -3 \] \[ x = -\frac{3}{2} \] Thus, the x-intercept is \( -\frac{3}{2} \).

Step 3: Find the y-intercept

To find the y-intercept, set the x-value to zero in the equation of the line, since at the y-intercept, the graph crosses the y-axis where x is zero. Solve the equation for y. Using the same example, set \( x = 0 \) and solve: \[ y = 2(0) + 3 \] \[ y = 3 \] So, the y-intercept is 3.

Key Concepts

x-intercept

In algebra, the x-intercept of a graph is a vital concept. It refers to the point where a graph intersects the x-axis. At this specific point, the value of the y-coordinate is zero. This location on a graph represents a solution where the line "crosses" or "touches" the x-axis.

To find the x-intercept algebraically, you will typically follow these steps:

• Set the y-value in the equation to zero.
• Solve the resulting equation for the x-value.

Let's use a simple example to illustrate this. Consider the equation of a line: \[ y = 2x + 3 \]To find the x-intercept:1. Replace \( y \) with zero, giving you: \[ 0 = 2x + 3 \]2. Solve for \( x \): \[ 2x = -3 \] \[ x = -\frac{3}{2} \]Thus, the x-intercept is the point \(-\frac{3}{2}, 0\). Understanding this concept facilitates solving linear equations and graphing them correctly.

y-intercept

The y-intercept is another crucial concept in algebra, defining the point where a graph crosses the y-axis. At this point, the value of the x-coordinate is zero. It's significant because it provides an initial starting point for graphing a line.

Here's how you find the y-intercept algebraically:

• Set the x-value in the equation to zero.
• Solve the modified equation for the y-value.

Let's consider the same line equation: \[ y = 2x + 3 \]To find the y-intercept, follow these steps:1. Plug in zero for \( x \): \[ y = 2(0) + 3 \]2. Simplify the equation to find \( y \): \[ y = 3 \]Therefore, the y-intercept is the point \(0, 3\). This point tells you where the graph can start when plotted and provides insight into the slope and position of the line.

graphing equations

Graphing equations is an essential skill to visualize algebraic relationships. By plotting points, particularly intercepts, you can understand how an equation behaves graphically. Intercepts serve as foundational points on the graph, allowing you to draw the line accurately.

Here's how you can graph an equation using intercepts:

• Calculate both the x-intercept and the y-intercept using the methods described earlier.
• Plot these intercepts on the coordinate plane.
• Once both intercepts are plotted, draw a straight line through them, extending it in both directions.

This simple process not only shows where the graph touches the axes but also reveals the slope of the line, indicating whether it increases or decreases across the graph. By focusing on intercepts, you gain a clearer insight into the graph's characteristics without needing to plot numerous points.

algebraic solutions

Algebraic solutions involve solving equations to find specific values, like intercepts, that provide meaningful information about a graph. Solving for x-intercepts and y-intercepts is part of this process, requiring manipulation of the equation.

Here's a brief overview of algebraic solutions concerning intercepts:

• Identify which part of the equation needs to be zero to find the specific intercept.
• Rearrange the equation algebraically to isolate the targeted variable.
• Solve for the intercept value.

For instance, when working with the equation \( y = 2x + 3 \), finding intercepts involves logical steps: 1. Set \( y = 0 \) to solve for the x-intercept. 2. Set \( x = 0 \) to solve for the y-intercept. This thorough understanding of algebraic solutions makes it easier to interpret and graph equations, providing clarity on mathematical relationships and their graphical representations.