Step into the realm of quadratic equations and let’s embark on a journey to visualize the enigmatic graph of y = 2x². This captivating curve holds secrets that will unfold before our very eyes, revealing its properties and behaviors. As we delve deeper into its characteristics, we’ll uncover its vertex, axis of symmetry, and the fascinating interplay between its shape and the quadratic equation that defines it. Brace yourself for a captivating exploration where the beauty of mathematics takes center stage.
To initiate our graphing adventure, we’ll begin by examining the equation itself. The coefficient of the x² term, which is 2 in this case, determines the overall shape of the parabola. A positive coefficient, like 2, indicates an upward-opening parabola, inviting us to visualize a graceful curve arching towards the sky. Moreover, the absence of a linear term (x) implies that the parabola’s axis of symmetry coincides with the y-axis, further shaping its symmetrical countenance.
As we continue our exploration, a crucial point emerges – the vertex. The vertex represents the parabola’s turning point, the coordinates where it changes direction from increasing to decreasing (or vice versa). To locate the vertex, we’ll employ a clever formula that yields the coordinates (h, k). In our case, with y = 2x², the vertex lies at the origin, (0, 0), a unique position where the parabola intersects the y-axis. This point serves as a pivotal reference for understanding the parabola’s behavior.
Plotting the Graph of Y = 2x^2
To graph the function Y = 2x^2, we can use the following steps:
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Create a table of values. Start by choosing a few values for x and calculating the corresponding values for y using the function Y = 2x^2. For example, you could choose x = -2, -1, 0, 1, and 2. The resulting table of values would be:
x y -2 8 -1 2 0 0 1 2 2 8 -
Plot the points. On a graph with x- and y-axes, plot the points from the table of values. Each point should have coordinates (x, y).
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Connect the points. Draw a smooth curve connecting the points. This curve represents the graph of the function Y = 2x^2.
Exploring the Equation’s Structure
The equation y = 2x2 is a quadratic equation, meaning that it has a parabolic shape. The coefficient of the x2 term, which is 2 in this case, determines the curvature of the parabola. A positive coefficient, as we have here, creates a parabola that opens upward, while a negative coefficient would create a parabola that opens downward.
The constant term, which is 0 in this case, determines the vertical displacement of the parabola. A positive constant term would shift the parabola up, while a negative constant term would shift it down.
The Number 2
The number 2 plays a significant role in the equation y = 2x2. It affects the following aspects of the graph:
| Property | Effect |
|---|---|
| Coefficient of x2 | Determines the curvature of the parabola, making it narrower or wider. |
| Vertical Displacement | Has no effect as the constant term is 0. |
| Vertex | Causes the vertex to be at the origin (0,0). |
| Axis of Symmetry | Makes the y-axis the axis of symmetry. |
| Range | Restricts the range of the function to non-negative values. |
In summary, the number 2 affects the curvature of the parabola and its position in the coordinate plane, contributing to its unique characteristics.
Understanding the Vertex and Axis of Symmetry
Every parabola has a vertex, which is the point where it changes direction. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
To find the vertex of y = 2x2, we can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic equation. In this case, a = 2 and b = 0, so the x-coordinate of the vertex is x = 0.
To find the y-coordinate of the vertex, we substitute this value back into the original equation: y = 2(0)2 = 0. Therefore, the vertex of y = 2x2 is the point (0, 0).
The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is 0, the axis of symmetry is the line x = 0.
| Vertex | Axis of Symmetry |
|---|---|
| (0, 0) | x = 0 |
Determining the Parabola’s Direction of Opening
The coefficient of x2 determines whether the parabola opens upwards or downwards. For the equation y = 2x2 + bx + c, the coefficient of x2 is positive (2). This means that the parabola will open upwards.
Table: Direction of Opening Based on Coefficient of x2
| Coefficient of x2 | Direction of Opening |
|---|---|
| Positive | Upwards |
| Negative | Downwards |
In this case, since the coefficient of x2 is 2, a positive value, the parabola y = 2x2 will open upwards. The graph will be an upward-facing parabola.
Creating the Graph Step-by-Step
1. Find the Vertex
The vertex of a parabola is the point where the graph changes direction. For the equation y = 2x2, the vertex is at the origin (0, 0).
2. Find the Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two equal halves. For the equation y = 2x2, the axis of symmetry is x = 0.
3. Find the Points on the Graph
To find points on the graph, you can plug in values for x and solve for y. For example, to find the point when x = 1, you would plug in x = 1 into the equation and get y = 2(1)2 = 2.
4. Plot the Points
Once you have found some points on the graph, you can plot them on a coordinate plane. The x-coordinate of each point is the value of x that you plugged into the equation, and the y-coordinate is the value of y that you got back.
5. Connect the Points
Finally, you can connect the points with a smooth curve. The curve should be a parabola opening upwards, since the coefficient of x2 is positive. The graph of y = 2x2 looks like this:
| x | y |
|---|---|
| -1 | 2 |
| 0 | 0 |
| 1 | 2 |
Calculating Key Points on the Graph
To graph the parabola y = 2x2, it’s helpful to calculate a few key points. Here’s how to do that:
Vertex
The vertex of a parabola is the point where it changes direction. For y = 2x2, the x-coordinate of the vertex is 0, since the coefficient of the x2 term is 2. To find the y-coordinate, substitute x = 0 into the equation:
| Vertex |
|---|
| (0, 0) |
Intercepts
The intercepts of a parabola are the points where it crosses the x-axis (y = 0) and the y-axis (x = 0).
x-intercepts: To find the x-intercepts, set y = 0 and solve for x:
| x-intercepts |
|---|
| (-∞, 0) and (∞, 0) |
y-intercept: To find the y-intercept, set x = 0 and solve for y:
| y-intercept |
|---|
| (0, 0) |
Additional Points
To get a better sense of the shape of the parabola, it’s helpful to calculate a few additional points. Choose any x-values and substitute them into the equation to find the corresponding y-values.
For example, when x = 1, y = 2. When x = -1, y = 2. These additional points help define the curve of the parabola more accurately.
Asymptotes
A vertical asymptote is a vertical line that the graph of a function approaches but never touches. A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity.
The graph of y = 2x2 has no vertical asymptotes because it is continuous for all real numbers. The graph does have a horizontal asymptote at y = 0 because as x approaches infinity or negative infinity, the value of y approaches 0.
Intercepts
An intercept is a point where the graph of a function crosses one of the axes. To find the x-intercepts, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y.
The graph of y = 2x2 passes through the origin, so the y-intercept is (0, 0). To find the x-intercepts, set y = 0 and solve for x:
$$0 = 2x^2$$
$$x^2 = 0$$
$$x = 0$$
Therefore, the graph of y = 2x2 has one x-intercept at (0, 0).
Transformations of the Parent Graph
The parent graph of y = 2x^2 is a parabola that opens upward and has its vertex at the origin. To graph any other equation of the form y = 2x^2 + k, where k is a constant, we need to apply the following transformations to the parent graph.
Vertical Translation
If k is positive, the graph will be translated k units upward. If k is negative, the graph will be translated k units downward.
Vertex
The vertex of the parabola will be at the point (0, k).
Axis of Symmetry
The axis of symmetry will be the vertical line x = 0.
Direction of Opening
The parabola will always open upward because the coefficient of x^2 is positive.
x-intercepts
To find the x-intercepts, we set y = 0 and solve for x:
0 = 2x^2 + k
x^2 = -k/2
x = ±√(-k/2)
y-intercept
To find the y-intercept, we set x = 0:
y = 2(0)^2 + k
y = k
Table of Transformations
The following table summarizes the transformations applied to the parent graph y = 2x^2 to obtain the graph of y = 2x^2 + k:
| Transformation | Effect |
|---|---|
| Vertical translation | The graph is translated k units upward if k is positive and k units downward if k is negative. |
| Vertex | The vertex of the parabola is at the point (0, k). |
| Axis of symmetry | The axis of symmetry is the vertical line x = 0. |
| Direction of opening | The parabola always opens upward because the coefficient of x^2 is positive. |
| x-intercepts | The x-intercepts are at the points (±√(-k/2), 0). |
| y-intercept | The y-intercept is at the point (0, k). |
Steps to Graph y = 2x^2:
1. Plot the Vertex: The vertex of a parabola in the form y = ax^2 + bx + c is (h, k) = (-b/2a, f(-b/2a)). For y = 2x^2, the vertex is (0, 0).
2. Find Two Points on the Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, which for y = 2x^2 is x = 0. Choose two points equidistant from the vertex, such as (-1, 2) and (1, 2).
3. Reflect and Connect: Reflect the points across the axis of symmetry to obtain two more points, such as (-2, 8) and (2, 8). Connect the four points with a smooth curve to form the parabola.
Applications in Real-World Scenarios
9. Projectile Motion: The trajectory of a projectile, such as a thrown ball or a fired bullet, can be modeled by a parabola. The vertical distance traveled, y, can be expressed as y = -16t^2 + vt^2, where t is the elapsed time and v is the initial vertical velocity.
To find the maximum height reached by the projectile, set -16t^2 + vt = 0 and solve for t. Substitute this value back into the original equation to determine the maximum height. This information can be used to calculate how far a projectile will travel or the time it takes to hit a target.
| Scenario | Equation |
|---|---|
| Trajectories of a projectile | y = -16t^2 + vt^2 |
| Vertical distance traveled by a thrown ball | y = -16t^2 + 5t^2 |
| Parabolic flight of a fired bullet | y = -16t^2 + 200t^2 |
Summary of Graphing Y = 2x^2
Graphing Y = 2x^2 involves plotting points that satisfy the equation. The graph is a parabola that opens upwards and has a vertex at (0, 0). The table below shows some of the key features of the graph:
| Point | Value |
|---|---|
| Vertex | (0, 0) |
| x-intercepts | None |
| y-intercept | 0 |
| Axis of symmetry | x = 0 |
10. Determining the Shape and Orientation of the Parabola
The coefficient of x^2 in the equation, which is 2 in this case, determines the shape and orientation of the parabola. Since the coefficient is positive, the parabola opens upwards. The larger the coefficient, the narrower the parabola will be. Conversely, if the coefficient were negative, the parabola would open downwards.
It’s important to note that the x-term in the equation does not affect the shape or orientation of the parabola. Instead, it shifts the parabola horizontally. A positive value for x will shift the parabola to the left, while a negative value will shift it to the right.
How to Graph Y = 2x^2
To graph the parabola, y = 2x^2, following steps can be followed:
- Identify the vertex: The vertex of the parabola is the lowest or highest point on the graph. For the given equation, the vertex is at the origin (0, 0).
- Plot the vertex: Mark the vertex on the coordinate plane.
- Find additional points: To determine the shape of the parabola, choose a few more points on either side of the vertex. For instance, (1, 2) and (-1, 2).
- Plot the points: Mark the additional points on the coordinate plane.
- Draw the parabola: Sketch a smooth curve through the plotted points. The parabola should be symmetrical about the vertex.
The resulting graph will be a U-shaped parabola that opens upward since the coefficient of x^2 is positive.
People Also Ask
What is the equation of the parabola with vertex at (0, 0) and opens upward?
The equation of a parabola with vertex at (0, 0) and opens upward is y = ax^2, where a is a positive constant. In this case, the equation is y = 2x^2.
How do you find the x-intercepts of y = 2x^2?
To find the x-intercepts, set y = 0 and solve for x. So, 0 = 2x^2. This gives x = 0. The parabola only touches the x-axis at the origin.
What is the y-intercept of y = 2x^2?
To find the y-intercept, set x = 0. So, y = 2(0)^2 = 0. The y-intercept is at (0, 0).
