Introduction
When looking at an equation such as Y = 1/2X^2, it is important to understand how it relates to graphing in mathematics. The equation represents a quadratic function, which is a type of polynomial function that can be graphed as a parabola. In this article, we will delve into how to graph Y = 1/2X^2 and determine which graph accurately represents this equation.
Understanding the Equation Y = 1/2X^2
Before we can determine which graph represents Y = 1/2X^2, it is essential to understand the components of the equation:
- Y: This represents the dependent variable, which is the output or result of the equation.
- X: This represents the independent variable, which is the input or given value in the equation.
- 1/2: This coefficient determines the stretch or compression of the parabola along the vertical axis (Y-axis).
- X^2: This term signifies that X is squared, indicating a quadratic relationship.
Graphing Y = 1/2X^2
To graph Y = 1/2X^2, we can follow these steps:
- Choose a set of values for X to create ordered pairs. For simplicity, we will use integer values such as -3, -2, -1, 0, 1, 2, and 3.
- Substitute the X values into the equation Y = 1/2X^2 to find the corresponding Y values.
- Plot the ordered pairs (X, Y) on a coordinate plane.
- Connect the points to create a smooth curve, representing the graph of Y = 1/2X^2.
Characteristics of Y = 1/2X^2
When graphing Y = 1/2X^2, several key characteristics should be noted:
- Vertex: The vertex of the parabola occurs at the point (0, 0) due to the coefficient of 1/2 shifting its position.
- Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, dividing the parabola into two equal halves.
- Direction: The parabola opens upwards because the coefficient of X^2 is positive.
- Shape: The shape of the parabola is wider or narrower depending on the value of the coefficient 1/2.
Identifying the Correct Graph
Now that we understand how to graph Y = 1/2X^2 and its characteristics, let’s analyze which visual representation accurately depicts this equation:
- Consider the Shape: Look for a parabolic curve that opens upwards and has a wider or narrower shape.
- Locate the Vertex: Identify the vertex of the parabola, which should be at the origin (0, 0).
- Check the Axis of Symmetry: Ensure that the graph exhibits symmetry about the vertical line passing through the vertex.
Conclusion
In conclusion, the equation Y = 1/2X^2 represents a quadratic function that can be graphed as a parabola. By understanding the components of the equation and its characteristics, we can accurately depict the graph of Y = 1/2X^2. By following the steps outlined in this article and analyzing the key features of the graph, we can determine which visual representation correctly represents the equation.
