Which Graph Represents Y 1 2 X 2

Which Graph Represents y = 1/2x²? A Deep Dive into Quadratic Functions and Their Visual Representations

Understanding the relationship between algebraic equations and their graphical representations is fundamental to mathematics. This article delves into the specific quadratic equation y = 1/2x², exploring its characteristics, how to graph it accurately, and how to distinguish it from other related quadratic functions. We'll also touch upon broader concepts within quadratic functions to provide a comprehensive understanding.

Understanding the Basic Quadratic Form

Before focusing on our specific equation, let's review the general form of a quadratic equation: y = ax² + bx + c. In this form:

a, b, and c are constants (real numbers).
a determines the parabola's vertical stretch or compression and its direction (opening upwards if a > 0, downwards if a < 0).
b affects the parabola's horizontal shift (or translation).
c represents the y-intercept (where the graph crosses the y-axis).

Our equation, y = 1/2x², is a simplified version of the general form. Here, a = 1/2, b = 0, and c = 0. This simplification allows us to easily identify key characteristics of the graph.

Key Characteristics of y = 1/2x²

Parabola Shape: Because the coefficient of x² (a) is positive (1/2), the parabola opens upwards. This means it has a U-shape.
Vertex: The vertex is the lowest point on the parabola since it opens upwards. In this case, because b = 0 and c = 0, the vertex is located at the origin, (0, 0).
Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola into two mirror images. It passes through the vertex. For our equation, the axis of symmetry is the y-axis (x = 0).
Vertical Stretch/Compression: The value of 'a' (1/2) indicates a vertical compression. Compared to the parent function y = x², the graph of y = 1/2x² is wider or less steep. This is because each y-value is half the corresponding y-value of y = x².
y-intercept: Since c = 0, the parabola intersects the y-axis at the origin, (0, 0).
x-intercept: The x-intercepts are the points where the graph crosses the x-axis (where y = 0). Setting y = 0 in the equation 1/2x² = 0, we find that x = 0. Therefore, the only x-intercept is at the origin, (0, 0).

Plotting the Graph: A Step-by-Step Guide

To accurately plot the graph of y = 1/2x², you can follow these steps:

Create a table of values: Choose several values for x (both positive and negative) and calculate the corresponding y-values using the equation y = 1/2x².

x	y = 1/2x²
-4	8
-3	4.5
-2	2
-1	0.5
0	0
1	0.5
2	2
3	4.5
4	8

Plot the points: On a Cartesian coordinate plane, plot the (x, y) points from your table.
Draw the parabola: Smoothly connect the plotted points to form a U-shaped curve. Remember the parabola is symmetrical about the y-axis.

Distinguishing y = 1/2x² from Other Quadratic Functions

It's crucial to understand how changing the values of a, b, and c in the general quadratic equation affects the graph. Let's compare y = 1/2x² with some variations:

y = x²: This is the parent function. Its graph is narrower than y = 1/2x².
y = 2x²: This graph is steeper than y = 1/2x² because of the larger coefficient of x².
y = 1/2x² + 2: This graph is identical to y = 1/2x² but shifted vertically upwards by 2 units. The vertex would be at (0, 2).
y = 1/2(x - 2)²: This graph is identical to y = 1/2x² but shifted 2 units to the right. The vertex would be at (2, 0).
y = -1/2x²: This graph is a reflection of y = 1/2x² across the x-axis. It opens downwards.

By understanding these variations, you can quickly identify which graph corresponds to a given quadratic equation.

Applications of Quadratic Functions

Quadratic functions have numerous applications in various fields:

Physics: Describing projectile motion (the trajectory of a ball, for example).
Engineering: Modeling the shape of parabolic antennas and bridges.
Economics: Representing cost, revenue, and profit functions.
Computer graphics: Creating curved shapes and animations.
Statistics: Fitting curves to data.

Understanding the graphical representation of quadratic functions, such as y = 1/2x², is essential for solving problems and interpreting results in these and other areas.

Advanced Concepts: Derivatives and Concavity

For a deeper understanding, we can introduce calculus. The first derivative of y = 1/2x², which is dy/dx = x, gives the slope of the tangent line at any point on the curve. The second derivative, d²y/dx² = 1, indicates the concavity of the function. Since the second derivative is positive, the parabola is concave up. This confirms our observation that the parabola opens upwards.

Conclusion

The graph of y = 1/2x² is a vertically compressed parabola opening upwards, with its vertex at the origin (0,0). Its symmetrical nature around the y-axis, and its relatively wider shape compared to the parent function y = x², are defining features. Understanding this specific function and the broader concepts of quadratic functions and their graphical representations is crucial for various mathematical and real-world applications. By carefully examining the coefficients in the general quadratic equation, you can accurately predict and visualize the shape and position of the resulting parabola. Remember to practice plotting different quadratic functions to solidify your understanding and develop your skills in analyzing their graphical representations. This will significantly enhance your comprehension of mathematical concepts and their application in diverse fields.