Problem 29 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$y=-x+4$$

Step-by-Step Solution

Answer

The graph of the equation \(y=-x+4\) is a line that crosses the y-axis at the point (0,4) and has a slope of -1.

Step 1: Identify the y-intercept

The y-intercept is the point where the line crosses the y-axis. Looking at the equation \(y=-x+4\), one can see that the y-intercept (represented by \(b\)) is 4. Therefore, the y-intercept is at the point (0,4). Plot this point on the graph.

Step 2: Identify the slope

The slope (represented by \(m\)) is -1. This means that for every 1 unit increase in the x-value, the y-value decreases by 1.

Step 3: Plot the line

Starting from the y-intercept (0,4), move down one unit and to the right one unit because of the slope of -1. This leads to the point (1,3). Plot this point on the graph. Repeat this process to get several points. After plotting, draw a line through the plotted points to represent the graph of \(y=-x+4\).

Key Concepts

Understanding the Y-intercept

The y-intercept is a crucial element in graphing linear equations. It is the point where the line crosses the y-axis, indicating the initial value of the function when the x-value is zero. In the equation \(y = -x + 4\), the y-intercept is represented by the constant term, which in this case is 4.

Therefore, the line crosses the y-axis at the point (0,4). This point is essential for plotting because it gives us a fixed starting point on the graph. When drawing any linear equation, identifying and plotting the y-intercept correctly ensures that your graph starts accurately.

• The y-intercept tells you where to begin drawing your line.
• It provides a clear point on the graph for plotting further points based on the slope.

Decoding the Slope

In the graph of a linear equation, the slope defines the steepness and direction of the line. The slope, often represented by \(m\) in the slope-intercept form \(y = mx + b\), shows the rate at which the y-value of a line changes relative to the x-value. In our equation \(y = -x + 4\), the slope \(m\) is -1. This indicates a downward trend because the slope is negative.

Here’s what the slope of -1 signifies:

• For every 1 unit increase in x, the y-value decreases by 1.
• This results in a line that slants downwards from left to right.
• A negative slope like -1 reflects that as x becomes larger, y becomes smaller.

Understanding the slope helps in predicting the direction in which the line will extend across the graph. It's a guiding measure for plotting additional points.

Strategies for Plotting Points

Plotting points is the next step after identifying the y-intercept and slope. To graph a line accurately, you start by plotting the y-intercept, then use the slope to determine the subsequent points on the line.

For the equation \(y = -x + 4\), we've established that the y-intercept is at (0,4). Start by marking this point on your graph. Then, use the slope to find other points:

• With a slope of -1, move down 1 unit vertically and right 1 unit horizontally from (0,4), reaching (1,3).
• Repeat this step to find more points like (2,2) or (3,1).

These points create a straight line when connected, which accurately depicts the linear relationship described by your equation. Plotting carefully ensures a precise representation of the line on the graph, making sure other points derived from the slope are correctly aligned.

Introduction to Linear Functions

Linear functions are one of the fundamental concepts in algebra and pre-calculus, characterized by their straight-line graphs. They are typically written in the form \(y = mx + b\), known as the slope-intercept form. This form explicitly shows where the line intercepts the y-axis and its slope.

Linear functions have key properties:

• Each point on the line satisfies the equation.
• They depict a constant rate of change, represented by the slope \(m\).
• Their simple structure allows them to be easily graphed and interpreted.

By understanding linear functions, you can easily predict the behavior of a model or scenario described by a linear equation. Such functions are prevalent in various fields, depicting real-world relationships where variables change linearly.