Problem 38 Use a table of values to graph t... [FREE SOLUTION]

Chapter 4: Problem 38

Use a table of values to graph the equation. \(y=-(3-x)\)

Short Answer

Expert verified

To graph the equation \(y=-(3-x)\) or \(y=x-3\), decide on a range of x-values, calculate the corresponding y-values, plot these points on the graph, and draw a line through the plotted points.

Step by step solution

Understand the equation

The provided linear equation is \(y=-(3-x)\). Now it can be rewritten as \(y=x-3\). Consequently, for every x, the y value will be equivalent to the x-value subtracted by 3.

Create a table of values

Begin by selecting several values for x and then calculate the corresponding y values using the equation \(y=x-3\). For example, if x is -1, 0, 1, 2, and 3, then the corresponding y values will be -4, -3, -2, -1, and 0, respectively.

Plot the table values on the graph

Having prepared a table of values, plot each (x,y) pair on the graph. Start by marking the points (-1,-4), (0,-3), (1,-2), (2,-1), (3,0) and continue for all pairs.

Draw the line

Once the points are marked, draw a line through them. This line is the graph of the equation \(y=x-3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Table of Values

A table of values is a systematic way to organize pairs of numbers that satisfy a given equation. To make one, select a range of values for the variable (commonly x). For each chosen x-value, compute the corresponding y-value using the equation at hand.
This approach makes it easy to visualize the relationship between x and y. In our exercise, we use the linear equation \(y = x - 3\).
Here, we've chosen x-values like -1, 0, 1, 2, and 3, and calculated the y-values as -4, -3, -2, -1, and 0, respectively.

It's key to choose a variety of x-values to see how y changes.
More values provide a clearer view of the line鈥檚 behavior.

Tables of values are helpful for beginners in graphing and provide clarity on how each pair of x and y relates according to the equation.

Linear Function

A linear function is a function that can be graphically represented in the coordinate plane as a straight line. Linear functions are often written in the form \(y = mx + b\), where m represents the slope and b indicates the y-intercept.
In our case, the function is \(y = x - 3\). Here, the slope m is 1, and the y-intercept b is -3, meaning that the line crosses the y-axis at -3.
Linear functions are characterized by a constant rate of change, this means for each unit increase in x, y changes by the slope.

Linear functions are simple yet fundamental in mathematics and applied in various fields such as economics and physics.
They help us understand relationships where one quantity depends directly and proportionally on another.

Always remember, the graph of a linear equation is a crucial visual tool that helps in understanding its behavior.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points and graph equations. The plane is defined by a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at a point called the origin.
Each point on the plane corresponds to a pair of values (x, y), where x is the horizontal position and y is the vertical position.
For our equation \(y = x - 3\), we find the coordinates for plotting such as (-1, -4), (0, -3), (1, -2), etc.

The coordinate plane is essential for graphing because it provides a reference to plot exact positions of numbers.
It helps in visually interpreting mathematical relationships and functions by showing the geometric shapes they create.

Mastering how to navigate the coordinate plane is fundamental for solving and graphing equations.

Plotting Points

Plotting points is the process of marking specific coordinates on a graph. Each point corresponds to a pair of numbers that satisfy the equation you are working with.
Begin by referencing your table of values. For each (x, y) pair, locate the x-value on the horizontal axis and the y-value on the vertical axis. Mark each intersection with a dot.
In the given example, points like (-1, -4), (0, -3), and (1, -2) are plotted successively.

Ensure accuracy when marking, as one misplaced point can lead to incorrect graphs.
After plotting, check that all points align before drawing the connecting line.

Plotting is a meticulous yet rewarding process that builds the foundation for visualizing and analyzing functions.

91影视

Use a table of values to graph the equation. \(y=-(3-x)\)

Short Answer

Step by step solution

Understand the equation

Create a table of values

Plot the table values on the graph

Draw the line

Key Concepts

Table of Values

Linear Function

Coordinate Plane

Plotting Points

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