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Magazine Chapter 1: Problem 1
Graph the given relation. $$ \\{(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9)\\} $$
Short Answer
Expert verified
Plot the given points on a coordinate plane to reveal a parabolic shape.
Step by step solution
01
Identify the Points
First, identify the given points from the relation set. These points form the basis of the graph: \((-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)\).
02
Prepare the Coordinate System
Set up a coordinate plane with a horizontal axis (x-axis) and a vertical axis (y-axis). Make sure the axes are numbered to cover the range of x-values from -3 to 3 and y-values from 0 to 9.
03
Plot the Points
Using the identified coordinates, plot each point on the graph. Mark the points: \((-3,9)\), \((-2,4)\), \((-1,1)\), \((0,0)\), \((1,1)\), \((2,4)\), and \((3,9)\).
04
Analyze the Point Pattern
Observe the pattern of the plotted points. Notice how the points are symmetrical on both sides of the y-axis and form a parabolic shape, suggesting a quadratic relationship.
05
Connect the Points (Optional)
If desired, connect the points with a smooth curve to help visualize the shape of the relation. This step confirms the parabolic nature of the relation, similar to a graph of a quadratic equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Plane
The coordinate plane is a two-dimensional plane formed by the intersection of a horizontal line known as the x-axis and a vertical line known as the y-axis. Together, these axes divide the plane into four regions called quadrants. This system allows us to locate points in a plane using ordered pairs of numbers.
When working with coordinate planes, it's important to understand:
When working with coordinate planes, it's important to understand:
- The x-axis runs horizontally and measures left and right movement.
- The y-axis runs vertically and measures up and down movement.
- The point where both axes intersect is called the origin, located at (0,0).
- Each point on the plane is represented as \( (x, y) \, \), where \( x \) is the horizontal value and \( y \) is the vertical value.
Plotting Points
Plotting points involves placing points on the coordinate plane using their respective coordinates. It's a straightforward process, once you understand how the x and y values guide position.
Here's how to plot points from a relation on a graph:
Here's how to plot points from a relation on a graph:
- Start at the origin (0,0), the center of the graph.
- Move horizontally to the x-value of the point. If the x-value is negative, move left. If positive, move right.
- Then, move vertically to the y-value. If the y-value is negative, move down. If positive, move up.
- Mark the point where these movements intersect with a dot.
Quadratic Relationship
A quadratic relationship occurs when points form a specific pattern described by a quadratic equation. Quadratic equations take the form \( ax^2 + bx + c = 0 \). They define a set of points that are proportional and exhibit symmetry.
Key features of quadratic relationships include:
Key features of quadratic relationships include:
- The relationship is expressed mathematically by a second-degree polynomial.
- The graph produced is symmetrical around a vertical axis, often the y-axis.
- Such relations often involve maximum or minimum values known as the vertex.
Parabolic Shape
A parabolic shape is what you get when graphing a quadratic relationship. It's characterized by a U or an inverted U curve, depending on the orientation of the parabola.
Here's more about parabolas:
Here's more about parabolas:
- A parabola opens upward if the x-square term coefficient is positive, like \( y = ax^2 \) with \( a > 0 \), resulting in a U shape.
- Conversely, it opens downward if the coefficient is negative, resulting in an inverted U shape.
- The point where the parabola changes direction is known as the vertex.
- The line that passes through the vertex and divides the parabola into mirror-image halves is known as the axis of symmetry.
