The equation y = 2x^2 is a fundamental concept in algebra and graphing, representing a parabola that opens upwards. This equation is a quadratic function, where the coefficient of x^2 is 2, indicating a vertical stretch of the basic parabola y = x^2. Understanding the graph of y = 2x^2 requires analyzing its key features, including the vertex, axis of symmetry, and x- and y-intercepts.
To begin, let's examine the general form of a quadratic equation: y = ax^2 + bx + c. In the case of y = 2x^2, the equation can be rewritten as y = 2x^2 + 0x + 0, where a = 2, b = 0, and c = 0. This means that the graph of y = 2x^2 is a vertical stretch of the basic parabola y = x^2 by a factor of 2.
The Vertex and Axis of Symmetry
The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)). For y = 2x^2, the vertex is (0, 0), since a = 2 and b = 0. The axis of symmetry is the vertical line passing through the vertex, which is x = 0.
This indicates that the graph of y = 2x^2 is symmetric about the y-axis. The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. For y = 2x^2, the vertex form is y = 2(x - 0)^2 + 0, which simplifies to y = 2x^2.
X- and Y-Intercepts
The x-intercepts of a parabola occur when y = 0. For y = 2x^2, this happens when 2x^2 = 0, which implies x = 0. Therefore, the only x-intercept is (0, 0).
The y-intercept occurs when x = 0. Substituting x = 0 into the equation y = 2x^2 yields y = 2(0)^2 = 0. Thus, the y-intercept is also (0, 0).
| Key Feature | Value |
|---|---|
| Vertex | (0, 0) |
| Axis of Symmetry | x = 0 |
| X-Intercept | (0, 0) |
| Y-Intercept | (0, 0) |
Key Points
- The equation y = 2x^2 represents a parabola that opens upwards.
- The vertex of the parabola is (0, 0), and the axis of symmetry is x = 0.
- The x- and y-intercepts are both (0, 0).
- The graph of y = 2x^2 is a vertical stretch of the basic parabola y = x^2 by a factor of 2.
- Understanding the graph of y = 2x^2 is crucial for analyzing more complex quadratic functions.
Graphing the Equation
To graph y = 2x^2, we can plot several points and connect them with a smooth curve. Since the vertex is (0, 0) and the axis of symmetry is x = 0, we can choose x-values on either side of 0 and calculate the corresponding y-values.
For example, when x = -1, y = 2(-1)^2 = 2. When x = 1, y = 2(1)^2 = 2. When x = -2, y = 2(-2)^2 = 8. When x = 2, y = 2(2)^2 = 8.
Plotting these points and connecting them with a smooth curve, we obtain the graph of y = 2x^2, which is a parabola that opens upwards and is symmetric about the y-axis.
Real-World Applications
The equation y = 2x^2 has numerous real-world applications, including physics, engineering, and economics. For instance, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation.
In conclusion, understanding the graph of y = 2x^2 is essential for working with quadratic functions in various fields. By analyzing its key features, including the vertex, axis of symmetry, and x- and y-intercepts, we can gain valuable insights into the behavior of more complex equations.
What is the vertex of the parabola y = 2x^2?
+The vertex of the parabola y = 2x^2 is (0, 0).
What is the axis of symmetry of the parabola y = 2x^2?
+The axis of symmetry of the parabola y = 2x^2 is x = 0.
What are the x- and y-intercepts of the parabola y = 2x^2?
+The x- and y-intercepts of the parabola y = 2x^2 are both (0, 0).