Problem 79 - Algebra A Combined Function

Question

Solve. See the Concept Checks in this section. Is the graph of (3,0) in the same location as the graph of (0,3) ? Explain why or why not.

Step-by-Step Solution

Answer

No, (3,0) and (0,3) are in different locations on the graph.

Step 1: Understand the Graphs Given

We have two points to consider: (3,0) and (0,3). These points are written in the format \((x, y)\), which denotes their position on a Cartesian coordinate plane.

Step 2: Locate the First Graph Point (3,0)

The point (3,0) indicates that it is located 3 units along the x-axis and 0 units along the y-axis. This means the point lies on the x-axis itself, specifically at x = 3.

Step 3: Locate the Second Graph Point (0,3)

The point (0,3) shows that it is located 0 units along the x-axis and 3 units along the y-axis. This means the point is on the y-axis, specifically at y = 3.

Step 4: Compare the Points

Since (3,0) is located on the x-axis and (0,3) is located on the y-axis, these points occupy different positions on the Cartesian plane. They are not in the same location.

Key Concepts

graphing points

When we talk about graphing points, we are referring to placing points on a Cartesian coordinate system. This system allows us to visualize the relationship between pairs of numbers. Each point can be visualized as a spot on a graph where its first number indicates its position horizontally, and the second number determines its position vertically.
The task of graphing involves identifying two numbers or coordinates. These numbers tell us exactly where to put the point on the graph. It's a straightforward step once you understand the basic concept.

• The first value in a pair of coordinates indicates how far to move along the horizontal line.
• The second value tells you how much to shift up or down vertically from the starting point.

By following these rules, you can create a clear visual understanding of mathematical relationships.

x-axis and y-axis

The x-axis and y-axis form the backbone of the Cartesian coordinate plane, running perpendicular to each other.
The x-axis is the horizontal line. It goes from left to right, representing all possible values of a point's first number or coordinate. This axis can extend both infinitely left and right, capturing both positive and negative values.
On the other hand, the y-axis stands vertically. It stretches upward and downward, capturing the second coordinate's values.

• Points on the x-axis have the form \(x, 0\) where the y-coordinate is zero.
• Points on the y-axis have the form \(0, y\) where the x-coordinate is zero.

Understanding these axes is essential for precisely locating points on the graph.

coordinates

Coordinates are the pairs of numbers used to precisely place points on a graph. Each pair is written as \(x, y\) where:\

• \(x\) is the horizontal position - the distance to move left or right from the origin.
• \(y\) is the vertical position - the measure of the shift up or down from the origin.

The origin, located at (0,0), is the point where these two axes intersect. From there, any point's position on the plane is determined by its coordinates.
Think of coordinates as instructions. They guide us on how far and in which direction to travel from the origin to find a point's position.
This \(x, y\) ordering is crucial since it keeps us consistent when graphing. Mixing them up would lead us to entirely different locations.