Question

Use a table of values to graph the equation. Label the \(x \text { -intercept and the } y \text { -intercept. (Review } 4.2,4.3)\) $$y=x+5$$

Short answer

The y-intercept is at (0,5) and the x-intercept is at (-5,0). After plotting these points, a line is drawn through them which represents the graph of the function \(y = x + 5\).

Step-by-step solution

Identify the equation

The given equation is \(y = x + 5\). This equation represents a linear function.

Find the y-intercept

To find the y-intercept, set \(x = 0\). Substituting \(x=0\) into the equation gets: \(y = 0 + 5\), so \(y=5\). Therefore, the y-intercept is at the point (0,5).

Find the x-intercept

To find the x-intercept, set \(y = 0\). Substituting \(y=0\) into the equation gets: \(0 = x + 5\). Solving for \(x\) gets \(x = -5\). Therefore, the x-intercept is at the point (-5,0).

Plot x-intercept and y-intercept

Plot the x-intercept and y-intercept on the graph. They should lie on the same straight line because we are plotting a linear function.

Draw the line

Draw a straight line through these two points. This line represents the graph of the function \(y=x+5\).

Key concepts

X-Intercept

Understanding the x-intercept of a linear equation is crucial to graphing linear equations. The x-intercept is the point where the graph of the equation crosses the x-axis. This happens when the y-value is zero. For a linear equation such as \(y = x + 5\), the x-intercept can be found by setting \(y=0\) and solving for \(x\). Doing so, we arrive at the equation \(0 = x + 5\) which simplifies to \(x = -5\). Therefore, the x-intercept is the point (-5,0).
This information allows you to start sketching your graph by marking this intercept on the x-axis, providing a reference point from which you can draw your linear function.

Y-Intercept

Similar to the x-intercept, the y-intercept is another important feature of linear functions. It represents the point where the graph intersects the y-axis, occurring when the x-value is zero. In our given equation \(y = x + 5\), the y-intercept is found by setting \(x=0\). This leads to the simplified form of \(y = 0 + 5\), which results in \(y=5\). Thus, the y-intercept is at (0,5).
The y-intercept provides the starting point on the y-axis for the graph of the linear function. By plotting this point, students can better visualize the slope of the line when connecting it to the previously found x-intercept.

Table of Values

A table of values is an essential tool for graphing a linear function. It consists of selected x-values and their corresponding y-values obtained from the equation of the line. To create such a table for the equation \(y = x + 5\), choose several values for \(x\) and calculate the corresponding \(y\) values. For instance:

• When \(x=-5\), \(y=0\) (the x-intercept).
• When \(x=0\), \(y=5\) (the y-intercept).
• When \(x=1\), \(y=1+5=6\).
• When \(x=2\), \(y=2+5=7\).

After plotting these values on the graph, draw a line through the points to represent the equation. By using a table of values, students gain a series of reference points which facilitate an accurate representation of the linear equation on a graph.

Linear Function

Linear functions, such as the one given by the equation \(y=x+5\), have graphs which are straight lines; this is a defining characteristic. The general form of a linear function is \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. For our specific function, the slope is 1 and the y-intercept is 5.
Understanding the concept of slope is important because it describes the steepness and direction of the line. The slope here tells us that for every one unit increase in \(x\), \(y\) increases by one unit as well. When graphing the function, the y-intercept provides the starting point, and the slope guides how we draw the line from one point to the next. It's this combination of slope and y-intercept that fully determines the graph of any linear function.