Problem 11 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=-x $$

Step-by-Step Solution

Answer

Both intercepts are at (0, 0). The graph is a line through the origin with slope -1.

Step 1: Understand the Equation

The given equation is \( y = -x \). This is a linear equation in slope-intercept form \( y = mx + b \), where the slope \( m = -1 \) and the y-intercept \( b = 0 \).

Step 2: Create a Table of Values

Choose a few values for \( x \) and compute the corresponding \( y \) values using the equation \( y = -x \). For example: When \( x = -2 \), \( y = -(-2) = 2 \); when \( x = 0 \), \( y = 0 \); when \( x = 2 \), \( y = -2 \).

Step 3: Plot the Points

Using the table of values from Step 2, plot the points on a coordinate plane. The points from the table include (-2, 2), (0, 0), and (2, -2).

Step 4: Draw the Line

Connect the plotted points in a straight line extending through each point. This line represents the graph of the equation \( y = -x \).

Step 5: Identify the Intercepts

For the y-intercept, set \( x = 0 \) in the equation to find \( y \): \( y = 0 \). This gives the y-intercept at the origin (0, 0). For the x-intercept, set \( y = 0 \) and solve for \( x \): \( 0 = -x \), which gives \( x = 0 \). Thus, the x-intercept is also at the origin (0, 0).

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and connecting them to form a line. It's a visual representation of relationships between variables. This can be understood through the equation provided, which is a linear function. In the example exercise, the equation given is \(y = -x\). This particular equation will create a line with a specific slope and intercept.

• First, you create a table of values. Select different values for \(x\) (such as -2, 0, and 2) and calculate the corresponding \(y\) values using the equation.
• For each \(x\), plug it into the equation to get \(y\). When \(x = -2\), \(y = 2\); when \(x = 0\), \(y = 0\); when \(x = 2\), \(y = -2\).
• These values form coordinate points (-2,2), (0,0), and (2,-2).

Once plotted, these points should lie on a straight line. Drawing a line through these plotted points shows the orientation of the graph determined by the linear equation \(y = -x\). This makes it apparent how changes in \(x\) impact \(y\) in a predictable manner across a linear space.

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations. It is represented as \(y = mx + b\) where \(m\) is the slope of the line and \(b\) is the y-intercept. Recognizing this form helps to quickly identify the slope and intercepts, crucial for graphing equations effectively. For the equation \(y = -x\), it is in slope-intercept form where the slope \(m\) is -1 and the y-intercept \(b\) is 0.

• The slope \(m = -1\) indicates that for every increase of 1 unit in \(x\), \(y\) decreases by 1 unit, suggesting a downward trend.
• The y-intercept \(b = 0\) implies that the line crosses the y-axis at the origin (0,0).

Combining the slope and y-intercept provides enough information to sketch the graph. Knowing this form greatly enhances understanding of a linear equation's behavior and its graphical representation. It simplifies the process of plotting linear equations and understanding their structure.

Intercepts

Intercepts are crucial points where the graph of an equation crosses the axes. Understanding intercepts helps in sketching graphs accurately. For linear equations, the x-intercept is where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.In the example of \(y = -x\), both the x-intercept and y-intercept occur at the origin (0,0).

• To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Here, setting \(x = 0\) gives \(y = 0\), identifying the y-intercept at (0,0).
• For the x-intercept, set \(y = 0\) and solve for \(x\). Solving \(0 = -x\) gives \(x = 0\), indicating the x-intercept is also at (0,0).

Intercepts like these are not only crucial identifying points on the graph but also provide insights into the nature of the line such as its direction and position relative to the axis. For \(y = -x\), both intercepts being at the same point simplifies understanding the linear equation's graph.