Problem 33 - Precalculus Essentials

Question

The table of values was generated by a graphing utility with a TABLE feature. Use the table to solve $$\begin{array}{|c|c|c|c|c|} \hline X & Y 1 & Y 2 & & \\ \hline-2 & 9 & 5 & & \\ -1 & 4 & 4 & & \\ 0 & 1 & 3 & & \\ 1 & 1 & 1 & & \\ 2 & 4 & 6 & & \\ 3 & 9 & -1 & & \\ 4 & 16 & -2 & & \\ 5 & 25 & -3 & & \\ 6 & 36 & -4 & & \\ \hline 7 & 49 & -5 & & \\ \hline X=-3 & & & & & \\ \hline \end{array}$$ Which equation corresponds to \(Y_{2}\) in the table? a. \(y_{2}=x+8\) b. \(\quad y_{2}=x-2\) c. \(y_{2}=2-x\) d. \(y_{2}=1-2 x\)

Step-by-Step Solution

Answer

The equation that corresponds to \(Y_{2}\) in the table is c. \(y_{2}=3-x\).

Step 1: Establish the changing pattern in Y2 column

Observing the tables, it can be seen that every time X increases by 1 (from -2 to 6), Y2 decreases by 1.

Step 2: Identify the y-intercept

The y-intercept is the value of Y2 when X equals 0. From the table, we can see that when X is 0, Y2 is 3.

Step 3: Write down the equation

So, for every increase of 1 in X, Y2 decreases by 1 (negative slope), and when X is 0, Y2 is 3 (y-intercept). This gives the equation \(y_{2}=3-x\).

Step 4: Match the equation with the provided answers

Comparing the equation with the provided answer choices, we can see that it matches with the equation provided in option c. So, the correct answer would be c. \(y_{2}=3-x\).

Key Concepts

Table of Values

Graphing linear equations often begins with creating a table of values. This is a tool that pairs each input, or x-value, of a function with its corresponding output, or y-value. To construct a table of values, you select a range of x-values and then calculate the corresponding y-values based on the function's formula.

Tables are exceptionally helpful because they provide a visual representation of how changes in x impact y. By examining patterns in the table, like constant increases or decreases, you can deduce characteristics of the function such as the slope, which represents the rate of change, or the y-intercept, which is the value where the line crosses the y-axis. We use this information to graph the line or to find the equation that correctly represents the relationship between x and y.

Y-Intercept

The y-intercept of a graph is a fundamental concept in the study of linear equations. It is the point where the line crosses the y-axis on a coordinate plane. To find the y-intercept from a table of values, look for the value of y when x is zero.

This specific value is crucial because it gives you a starting point for graphing the line and is part of the standard equation of a line. In the given exercise, the y-intercept is identified by observing the value of y associated with x=0, which is used subsequently to help determine the correct equation for Y_{2}. Knowing the y-intercept allows for a more accurate and easier graph plotting.

Slope

The slope of a line indicates its steepness and direction. Mathematically, it is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. For a linear equation in the form y = mx + b, m represents the slope. A positive slope means the line inclines upwards as it moves from left to right, while a negative slope means the line declines.

In the exercise, the slope was determined by observing how Y_{2} changes as x increases by 1. The consistent decrease indicated a negative slope. Recognizing the slope helped in writing the equation of the line by understanding how the y-value changes in relation to x.

Equation of a Line

The equation of a line is an algebraic representation of all the points that make up that line on a coordinate plane. The most common forms are the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, and the standard form (Ax + By = C).

When you have the slope and the y-intercept, as we found from our table of values, you can write the line's equation. In our example, the exercise showed that the line associated with Y_{2} decreases by one unit for every one unit increase in x, which we expressed as a negative slope, and it crosses the y-axis at 3. Thus, the equation fitting these observations is y = 3 - x, corresponding to option 'c.' Understanding how to derive and interpret the equation of a line is essential for graphing linear equations and analyzing their characteristics.