Problem 81 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. \(y=-(x-4)\)

Step-by-Step Solution

Answer

Generate a table of values with selected x-values and calculated y-values, then plot these ordered pairs on a Cartesian plane and draw a straight line that goes through them to represent the graph of the equation \(y=-(x-4)\)

Step 1: Create a Table of Values

Create a table that includes several x-values, starting from a negative number through zero to a positive number. A common selection would be x-values like -2,-1,0,1,2,3,4 and 5. The need for these values is to cover a wide spectrum of the graph.

Step 2: Calculate Corresponding Y-Values

Substitute each x-value into the given equation \(y=-(x-4)\) and compute the corresponding y-values. The generated y-values and the corresponding x-values form ordered pairs (x, y) which will be plotted on the graph.

Step 3: Plot the Points and Draw the Line

Using the ordered pairs from the table of values, plot each point on the Cartesian plane. The x-value is plotted along the horizontal axis, and the y-value is plotted on the vertical axis. After all points are plotted, draw a straight line that goes through these points. This line represents the graph of the equation

Key Concepts

Table of Values

When graphing equations, creating a table of values is a helpful first step. This method involves selecting various x-values and computing their corresponding y-values.
The goal is to find enough points to accurately represent the graph. For instance, in the equation given, \( y = -(x-4) \), choosing values for \( x \) such as -2, -1, 0, 1, 2, 3, 4, and 5 can provide a diverse range spanning both negative and positive domains.
Each selected \( x \) value is substituted back into the equation to find \( y \).

• When \( x = -2 \), replace it in the equation: \( y = -(-2-4) = 6 \)
• Repeat for other values: \( x = 0 \) gives \( y = 4 \), \( x = 4 \) gives \( y = 0 \), etc.

This approach not only prepares you for graphing but also helps in understanding the relationship between the two variables.

Ordered Pairs

Ordered pairs are essential to graphing on the Cartesian plane. These pairs (like \((x, y)\)) arise from your table of values and help us plot points precisely.
In the context of the given equation, each calculated \((x, y)\) forms an ordered pair, such as\((-2, 6)\), \((0, 4)\), and \((4, 0)\).
These pairs follow a format where the first number is the x-coordinate, indicating horizontal placement, and the second is the y-coordinate, indicating vertical placement.
Working with ordered pairs:

• Each coordinate pair is plotted based on their specific x and y values.
• Coordinates reflect the intersection of their respective x and y values on the plane.

The clarity of this two-number system simplifies the study of how changes in \(x\) influence \(y\). This understanding is central in interpreting graphs of functions and equations.

Cartesian Plane

The Cartesian plane is a two-dimensional grid set by perpendicular number lines, the x-axis (horizontal) and y-axis (vertical). It is here that ordered pairs are plotted to form graphs.
Each point plotted from ordered pairs corresponds exactly to an intersection of an x-value and a y-value on this plane.
Let's understand the role of the Cartesian plane further:

• The x-axis and y-axis divide the plane into four quadrants, each allowing visualization of positive and negative values of both axes.
• Graphing \( y = -(x-4) \) involves moving vertically for y-values according to their corresponding x-values as derived from the equation.

By plotting and connecting points like \((0, 4)\), \((4, 0)\), and more, you form a line representing the equation on this plane. This approach assists in visualizing relationships within an equation, aiding in conceptual comprehension.