Problem 29 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The graph of the equation y = -x is a straight line passing through the origin (0,0) and has a slope of -1.

Step 1: Identify the Slope and Y-intercept

For the equation of the form \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept. In this case, the equation is \(y = -x\) which can be rewritten as \(y = -x + 0\). So the slope \(m = -1\) and the y-intercept \(b = 0\).

Step 2: Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis. Since \(b = 0\), the line crosses the y-axis at the origin (0,0). Mark this point on the graph.

Step 3: Use the Slope to Find Another Point

The slope is the rise over the run, which is -1 in this case. This means for every 1 unit we move to the right along the x-axis, we move 1 unit down along the y-axis. From the y-intercept (0,0), move to the point (1,-1) and mark this point on the graph.

Step 4: Draw the Line

Now we have two points, (0,0) and (1,-1). Draw a line that passes through these two points. This line is the graph of the equation \(y = -x\).

Key Concepts

Slope

When discussing linear equations, the idea of slope is crucial to understand. The slope tells us how steep or flat the line is and determines the direction in which the line moves. In a linear equation represented by the form \(y = mx + b\), the letter \(m\) stands for the slope. This slope is a key value indicating how many units a line rises or falls when moving horizontally across the graph. Let's break it down:

• If the slope is positive, the line moves upwards from left to right.
• If the slope is negative, like in the equation \(y = -x\), the line moves downwards from left to right.
• A slope of 0 would make the line horizontal, showing no vertical change.

Now, why is the slope termed as 'rise over run'? Because it's expressed as a fraction outlining the vertical change (rise) over horizontal change (run) between any two points on the line. For the equation \(y = -x\), the slope is \(-1\), which means for every unit you move horizontally to the right, the line moves down 1 unit vertically. Understanding the slope helps you anticipate the behavior and position of the line on a graph.

Y-intercept

The y-intercept is another fundamental component when graphing linear equations. It tells us where the line crosses the y-axis. Such knowledge is pivotal in outlining the starting point of a graph.In the typical linear equation form \(y = mx + b\), the y-intercept is represented by the constant \(b\). This term signifies the point on the y-axis, allowing for quick identification of where the line begins on the graph.Here's how it works:

• The y-intercept is the value of \(y\) when \(x\) is 0.
• In our equation \(y = -x\) rewritten as \(y = -x + 0\), the y-intercept is 0, so the line crosses the y-axis at the origin (0,0).
• Every line graph will touch the y-axis at this specific point.

Recognizing the y-intercept allows you to effortlessly find a starting point on a graph. From there, using the slope, you can continue and plot additional points to complete the graph.

Coordinate Plane

The coordinate plane is like a map for graphing, helping us visualize equations clearly. It's composed of two axes that intersect at a point called the origin.The axes:

• x-axis: A horizontal line where values can be positive to the right of origin and negative to the left.
• y-axis: A vertical line where values can be positive above the origin and negative below.
• The origin: The center point of the plane located at (0,0), where these axes intersect.

Each point on the plane can be expressed as a pair of numbers called coordinates, often written as \((x, y)\). These coordinates help in pinpointing exact locations on the graph.As you graph a linear equation such as \(y = -x\), start by marking the y-intercept on the coordinate plane—in this case, the origin. Using the plane enables you to track and draw the slope correctly to find other crucial points, like moving from (0,0) to (1,-1).Understanding how to interpret and use the coordinate plane is vital for plotting linear graphs accurately and effortlessly.