Question

Use a table of values to graph the equation. Label the \(x \text { -intercept and the } y \text { -intercept. (Review } 4.2,4.3)\) $$y=-x-1$$

Short answer

The \(x\)-intercept of the graph is at (-1,0) and the \(y\)-intercept is at (0,-1). The equation is correctly represented by a straight line moving downward from left to right. All points on the graph adhere to the equation \(y = -x -1\).

Step-by-step solution

Creating a table of values

Choose values for \(x\), then determine the corresponding values for \(y\) using the equation \(y=-x-1\). For example, the table of values could look like: \[-2, -1, 0, 1, 2\] for \(x\) and \[1, 0, -1, -2, -3\] for \(y\].

Plot the points on a graph

Plot the points from the table on a graph. The \(x\) values represent the horizontal position and the \(y\) values represent the vertical position. The points are (-2,1), (-1,0), (0,-1), (1,-2), and (2,-3).

Drawing the line

Draw a straight line through the points. If the plotted points appear to form a straight line, then your table of values is accurate.

Labeling the intercepts

The \(x\)-intercept is the point where the line crosses the \(x\) axis, in this case it is (-1,0). The \(y\)-intercept is the point where the line crosses the \(y\) axis, in this case it is (0,-1). Label these points.

Key concepts

Table of Values

When graphing a linear equation like \(y = -x - 1\), a table of values is a helpful tool to map out the relationship between the two variables, \(x\) and \(y\). To create a table of values, select several values for \(x\), and compute the corresponding \(y\) values using the equation.

Start with simple values for \(x\), usually including zero, positive and negative numbers. For example, you might choose \(-2, -1, 0, 1, 2\). After substituting these \(x\) values into the equation, you get your \(y\) values, which could be \(1, 0, -1, -2, -3\) respectively. Organizing these pairs into a table gives you specific points that you can plot on a coordinate graph.

X-Intercept

The \(x\)-intercept of a graph is the point where the line crosses the \(x\)-axis. This event occurs when the \(y\)-value is zero. In the equation \(y = -x -1\), you can find the \(x\)-intercept by setting \(y\) equal to zero and solving for \(x\).

The equation becomes \(0 = -x - 1\), and solving it gives \(x = -1\). Thus, the \(x\)-intercept is the point \((-1, 0)\). It's important to identify this point in a graph because it gives information about where the line will cross the horizontal axis of the plane.

Y-Intercept

Conversely, the \(y\)-intercept is where the line crosses the \(y\)-axis, which happens when \(x\) is zero. Returning to our equation \(y = -x -1\) and plugging in \(x = 0\) gives us \(y = -0 - 1\), simplifying to \(y = -1\).

The \(y\)-intercept here is \((0, -1)\). It's an essential characteristic of the graph because it represents the initial value of \(y\) when \(x\) is just beginning, from which the line extends across the coordinate plane.

Coordinate Plotting

Coordinate plotting is the process of representing the table of values as points on a graph. The \(x\)-value dictates the horizontal position while the \(y\)-value determines the vertical position. For the equation \(y = -x - 1\), you would plot the points \((-2, 1)\), \((-1, 0)\), \((0, -1)\), \((1, -2)\), and \((2, -3)\).

After plotting these points on a Cartesian coordinate system, a pattern emerges that should form a straight line for a linear equation. You then draw a line through the plotted points to represent the equation visually on the graph. Be sure to extend the line to the edges of the graph and beyond the plotted points to cover all potential \(x\)-values.