Question

Group Activity In Exercises \(33-36,\) copy and complete the table for the function. $$y=-3 x+4$$

Short answer

The table will look similar to this: | x | y | |:-:|:-:| | -2 | 10 | | -1 | 7 | | 0 | 4 | | 1 | 1 | | 2 | -2 |

Step-by-step solution

Determine the table structure

The table typically will have two columns. The first column will include a range of x-values. The second column will be left blank, these will be filled in with the corresponding y-values that result from substituting each x-value into the equation.

Select and substitute x-values into the equation

Choose x-values. These can be any numeric values but for simplicity sake, we will use -2, -1, 0, 1 and 2 as our x-values. For each x-value substitute it into the equation. For example, when x is 0, \(y = -3(0) + 4 = 4\). Continue this process for each chosen x-value.

Fill in the table

List down the chosen x-values in the first column of the table. In the second column, write the corresponding y-value next to each x-value.

Key concepts

Understanding Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form of y = mx + b. In this representation, m stands for the slope of the line, b represents the y-intercept, and x and y are variables. The slope defines how steep the line is, and the y-intercept is the point where the line crosses the y-axis.

To understand how a linear equation forms a straight line when graphed, let's consider the given example in the exercise: y = -3x + 4. Here, the slope (m) is -3, and the y-intercept (b) is 4. This means that for every one unit increase in x, the value of y decreases by three units, and when x is zero, y will be four.

Substituting x-values

Substituting x-values into a linear equation is a straightforward process that allows us to find the corresponding y-value for each x-value. To properly fill out a function table, as the original exercise indicates, students should choose a range of x-values and then use the equation to find the y-values.

For the equation y = -3x + 4, you might select x-values like -2, -1, 0, 1, and 2. Take each of these values and plug them into the equation instead of x. For instance, if you substitute -1 for x, the equation becomes y = -3(-1) + 4 which simplifies to y = 7. Perform this operation for each of your chosen x-values to fill the corresponding y-values into the function table. This approach allows for an understanding of the function behavior by providing a snapshot of specific (x, y) pairs.

Graphing Linear Functions

Graphing linear functions involves plotting (x, y) pairs on a coordinate plane and then drawing a line through them, which visually represents the relationship defined by the linear equation. After completing the function table by substituting x-values into the equation, you can use those (x, y) pairs to graph the function.

For example, using the function y = -3x + 4 and the given x-values, you will have a set of coordinates, such as (0, 4), (1, 1), and (2, -2). Plot these points on a graph with x-axis and y-axis. Make sure each point corresponds correctly to its x-value (horizontal position) and y-value (vertical position). Once all the points are plotted, draw a straight line through them, and extend this line in both directions. This line represents the graph of the linear function and offers a visual way to predict y-values for any given x-value within the domain.