Problem 82 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. \(y=\frac{1}{2} x-5\)

Step-by-Step Solution

Answer

The graph is a straight line with a slope of 1/2 and y-intercept at -5. The line passes through the points (-2,-6), (-1,-5.5), (0,-5), (1,-4.5), and (2, -4).

Step 1: Choose values for x

Pick five values for x to create your table. A good choice would be -2, -1, 0, 1, and 2.

Step 2: Calculate corresponding y-values

With chosen x-values plug them in into the equation to find y-values. For example, for x=0, \(y=0.5*0-5=-5\).

Step 3: Complete the table

Continue to calculate the y-values for all other x-values. Now, the table will look like this: x= -2, y= -6; x= -1, y= -5.5; x=0, y= -5; x=1, y= -4.5; x=2, y= -4;

Step 4: Plot the points on the graph

Plot your points on the graph. They should form a straight line, which represents the equation \(y=\frac{1}{2} x-5\).

Key Concepts

Table of Values

When graphing linear equations, a 'table of values' is a powerful tool for creating a visual representation of the relationship between two variables. The process involves selecting input values, plugging them into the equation, and calculating the corresponding output.

For example, with the equation \(y=\frac{1}{2}x-5\), you start by choosing a set of 'x' values, such as -2, -1, 0, 1, and 2. You then calculate the 'y' values by substituting each 'x' into the equation.

Example Calculation

For an 'x' value of 1, the 'y' value would be \(\frac{1}{2}*1 - 5 = -4.5\). Repeat this for each chosen 'x' value and record the pairs in the table. The complete table is an organized way to view the pairs before graphing them.

Linear Equation

A 'linear equation' forms the foundation of coordinate graphing. It represents a straight line and can be written in various forms, with the most common being \(y=mx+b\), where 'm' is the slope and 'b' is the y-intercept.

Graphing the equation \(y=\frac{1}{2}x-5\), the slope is \(\frac{1}{2}\), showing that for each step taken to the right along the x-axis, the value on the y-axis increases by 0.5. The y-intercept is -5, indicating where the line crosses the y-axis. Understanding the slope and intercept provides insight into the behavior of the line and aids in graphing.

Coordinate Graphing

Using the table of values previously calculated, 'coordinate graphing' is the next step. It involves marking the (x, y) points on a graph and connecting them to reveal the line represented by the linear equation.

Starting with the point from our table, such as (-2, -6), place a dot on the graph where x is -2 and y is -6. Continue with the remaining points from the table. Once all points are on the graph, draw a line through them.

Consistency Check

Ensure that these points form a straight line - a crucial check for the correct plotting of a linear equation. This visual depiction aids in understanding the relationship between the variables and the equation's impact on their values.