Are you a math enthusiast eager to delve into the captivating world of functions and graphing? If so, let’s embark on an intriguing journey to unlock the secrets of graphing the enigmatic equation y = 1/2x². This quadratic function exhibits a distinctive parabolic shape that conceals hidden patterns and valuable insights. Join us as we unravel the intricacies of this mathematical masterpiece, exploring its graph’s characteristics, key features, and the steps involved in constructing its visual representation.
The graph of y = 1/2x² is a parabola that opens upward, inviting us to investigate its graceful curvature. Unlike linear functions, which follow a straight path, this parabola exhibits a symmetric arch, reaching its minimum point at the vertex. This key feature serves as the parabola’s focal point, where it transitions from decreasing to increasing values. Furthermore, the parabola’s axis of symmetry, a vertical line passing through the vertex, acts as a mirror line, reflecting each point on one side of the axis to a corresponding point on the other.
To unveil the graph of y = 1/2x², we must meticulously plot its points. Commence by selecting a series of x-values and calculating their corresponding y-values using the equation. These points will serve as building blocks for the parabola’s skeleton. As you plot these points, pay attention to the shape emerging before you. Gradually, the parabolic curve will take form, revealing its distinct characteristics. Remember, accuracy is paramount in this endeavor, ensuring that your graph faithfully represents the underlying function.
Understanding the Concept of a Parabola
Parabolas are U-shaped curves that are formed by the intersection of a cone with a plane parallel to its side. They have a vertex, which is the lowest point of the parabola, and a focus, which is a fixed point that determines the shape of the parabola. The equation of a parabola is generally given in the form y = ax^2 + bx + c, where a, b, and c are constants. The value of "a" determines the overall shape and orientation of the parabola. A positive value of "a" indicates that the parabola opens upward, while a negative value of "a" indicates that the parabola opens downward. The larger the absolute value of "a," the narrower the parabola.
Properties of Parabolas
Parabolas have several key properties that are important to understand when graphing them:
- Symmetry: Parabolas are symmetric about their axis of symmetry, which is a vertical line passing through the vertex.
- Vertex: The vertex is the lowest or highest point of the parabola and is located at x = -b/2a.
- Focus: The focus is a fixed point that determines the shape of the parabola. It is located at (0, 1/4a) for parabolas that open upward and (0, -1/4a) for parabolas that open downward.
- Directrix: The directrix is a horizontal line that is perpendicular to the axis of symmetry and is located at y = -1/4a for parabolas that open upward and y = 1/4a for parabolas that open downward.
Graphing Parabolas
To graph a parabola, you need to first identify the vertex, focus, and directrix. The vertex is the point where the parabola changes direction. The focus is the point that the parabola is reflecting off of. The directrix is the line that the parabola is opening up to. Once you have identified these three points, you can plot them on a graph and draw the parabola.
Plotting the Vertex
The vertex of a parabola is the point where it changes direction. To find the vertex of the parabola y = 1/2x^2, we need to use the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. In this case, a = 1/2 and b = 0, so the x-coordinate of the vertex is x = 0.
To find the y-coordinate of the vertex, we plug x = 0 back into the equation: y = 1/2(0)^2 = 0. Therefore, the vertex of the parabola y = 1/2x^2 is at the point (0,0).
Finding the Axis of Symmetry
The axis of symmetry of a parabola is a vertical line that passes through the vertex. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry is x = 0.
Determining the Opening of the Parabola
The opening of a parabola is the direction in which it opens. If the coefficient of the x^2 term is positive, the parabola opens upward. If the coefficient of the x^2 term is negative, the parabola opens downward. In this case, the coefficient of the x^2 term is positive, so the parabola y = 1/2x^2 opens upward.
Creating a Table of Values
To graph the parabola, we can create a table of values. We choose several x-values and calculate the corresponding y-values.
| x | y |
|—|—|—|
| -3 | 4.5 |
| -2 | 2 |
| -1 | 0.5 |
| 0 | 0 |
| 1 | 0.5 |
| 2 | 2 |
| 3 | 4.5 |
Finding the Intercepts
To determine the intercepts, substitute \(y = 0\) and \(x = 0\) into the equation, respectively.
y-intercept
Substitute \(y = 0\) into \(y = 1/2x^2\):
0 = 1/2x^2 x^2 = 0 x = 0
The y-intercept is \( (0,0) \).
x-intercepts
Substitute \(x = 0\) into \(y = 1/2x^2\):
y = 1/2(0)^2 y = 1/2(0) y = 0
Since \(y\) is always 0 when \(x = 0\), there are no x-intercepts.
Determining the Direction of Opening
The coefficient of the squared term, a, determines the direction of opening of the parabola:
- If a > 0, the parabola opens upward.
- If a < 0, the parabola opens downward.
In your case, for the equation y = \frac{1}{2}x^2, since a = \frac{1}{2} > 0, the parabola opens upward.
Additionally, you can verify the direction of opening by examining the vertex, which is the point where the parabola changes direction. The vertex form of a parabola is given by:
y = a(x – h)^2 + k,
where (h, k) is the vertex of the parabola.
By comparing the given equation with the vertex form, you can identify the coefficient a as \frac{1}{2}, which is positive. This further confirms that the parabola opens upward.
Graphing y = 1/2x2
Completing the Square (Optional)
Completing the square is an advanced technique that can be used to graph quadratic functions. For the function y = 1/2x2, we can complete the square as follows:
- Divide both sides of the equation by 1/2:
- Add (1/4) to both sides of the equation:
- Factor the left side of the equation:
- Divide both sides of the equation by 2:
- Subtract 1/4 from both sides of the equation:
2y = x2
2y + (1/4) = x2 + (1/4)
2(y + 1/4) = (x + 0)2
y + 1/4 = (x + 0)2/2
y = (x + 0)2/2 – 1/4
The equation y = (x + 0)2/2 – 1/4 is now in vertex form, which makes it easy to graph. The vertex of the parabola is at (0, -1/4), and the parabola opens upward.
Finding the x-Intercepts
To find the x-intercepts, we set y = 0 and solve for x:
0 = 1/2x2
x = 0
Therefore, the x-intercepts are (0, 0).
Finding the y-Intercept
To find the y-intercept, we set x = 0 and solve for y:
y = 1/2(0)2
y = 0
Therefore, the y-intercept is (0, 0).
Creating a Table of Values
To create a table of values, we choose several values of x and calculate the corresponding values of y:
| x | y |
|---|---|
| -2 | 2 |
| -1 | 1/2 |
| 0 | 0 |
| 1 | 1/2 |
| 2 | 2 |
Sketching the Graph
Using the information we have gathered, we can now sketch the graph of y = 1/2x2:
- Plot the vertex (0, -1/4).
- Plot the x- and y-intercepts (0, 0).
- Draw a smooth curve through the three points.
The graph of y = 1/2x2 is a parabola that opens upward and has its vertex at (0, -1/4).
Using a Table of Values
To graph the equation y = 1/2x², a table of values can be useful. This involves assigning values to x, calculating the corresponding y-values, and plotting the points. A table is a systematic way to organize these values.
Steps for Creating a Table of Values:
- Choose x-values: Select a range of x-values that will provide a good representation of the graph. Include both positive and negative values, if possible.
- Calculate y-values: For each x-value, square it (x²) and then divide the result by 2. This will give you the corresponding y-value.
- Create a table: Create a table with three columns: x, x², and y.
- Fill in the table: Enter the chosen x-values, their squared values, and the calculated y-values.
Example Table:
| x | x² | y |
|---|---|---|
| -2 | 4 | 2 |
| -1 | 1 | 0.5 |
| 0 | 0 | 0 |
| 1 | 1 | 0.5 |
| 2 | 4 | 2 |
Using the Table to Graph:
Once the table is complete, you can plot the points from the table on a graph.
- Label the axes: Label the horizontal axis as "x" and the vertical axis as "y".
- Plot the points: Mark the points from the table on the graph using a pencil or pen.
- Connect the points: Draw a smooth curve through the points to create the graph of the equation y = 1/2x².
By using a table of values, you can accurately plot the graph of a quadratic equation like y = 1/2x². This systematic approach helps ensure precision and provides a clear visual representation of the equation’s behavior.
7. Finding the Vertex and Axis of Symmetry
The vertex of a parabola is its turning point. To find the vertex of y = 1/2x^2, complete the square:
1/2x^2 = 1/8(2x^2) + 0 = 1/8(2x^2 – 8x + 16 – 16) + 0
1/2x^2 = 1/8(2x – 4)^2 – 2
Thus, the vertex is (2, -2).
The axis of symmetry is a vertical line passing through the vertex. The axis of symmetry for y = 1/2x^2 is x = 2.
| Step | Calculation |
|---|---|
| 1 | Subtract b2/4a (4 for this case) from x2. |
| 2 | Factor the resulting expression, taking out 1/4a (1/8 for this case) from (x ± b/2a)2. |
| 3 | Add 1/4a (2 for this case) back to the right of the equation to maintain equality. |
| 4 | Simplify the expression to find the vertex (h, k). |
Labeling the Axes
The first step in graphing a quadratic equation is to label the axes. The x-axis is the horizontal line that runs from left to right, and the y-axis is the vertical line that runs from bottom to top. The point where the two axes intersect is called the origin.
To label the axes, we need to choose a scale for each axis. This will determine how many units each line on the graph represents. For example, we might choose a scale of 1 unit per line for the x-axis and 2 units per line for the y-axis.
Once we have chosen a scale, we can label the axes. We start by labeling the origin as (0, 0). Then, we move along the x-axis in increments of our chosen scale and label the lines accordingly. For example, if we have chosen a scale of 1 unit per line, then we would label the lines as -3, -2, -1, 0, 1, 2, 3, and so on.
We do the same thing for the y-axis, but we start by labeling the origin as (0, 0) and move along the axis in increments of our chosen scale. For example, if we have chosen a scale of 2 units per line, then we would label the lines as -6, -4, -2, 0, 2, 4, 6, and so on.
| X-Axis | Y-Axis |
|---|---|
| -3 | -6 |
| -2 | -4 |
| -1 | -2 |
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
Adding Additional Information (e.g., intercepts, equation)
To further enhance the graph, you can add additional information such as intercepts and the equation of the parabola:
Intercepts
The x-intercepts are the points where the parabola crosses the x-axis. To find these points, set y to 0 in the equation and solve for x:
“`
0 = 1/2x^2
x = 0
“`
Therefore, the x-intercepts are (0, 0).
The y-intercept is the point where the parabola crosses the y-axis. To find this point, set x to 0 in the equation and solve for y:
“`
y = 1/2(0)^2
y = 0
“`
Therefore, the y-intercept is (0, 0).
Equation
The equation of the parabola can be written in the general form:
“`
y = ax^2 + bx + c
“`
For the parabola defined by y = 1/2x^2, the values of a, b, and c are:
| a | b | c |
|---|---|---|
| 1/2 | 0 | 0 |
Therefore, the equation of the parabola is:
“`
y = 1/2x^2
“`
Analyzing the Graph (e.g., vertex, axis of symmetry)
The graph of y = -1/2x2 is a parabola that opens downward. Its vertex is located at the origin (0, 0), and its axis of symmetry is the y-axis.
Vertex
The vertex of a parabola is the point where the parabola changes direction. The vertex of y = -1/2x2 is located at (0, 0). This is because the coefficient of x2 is negative, which indicates that the parabola opens downward. As a result, the vertex is the highest point on the parabola.
Axis of Symmetry
The axis of symmetry of a parabola is the vertical line that passes through the vertex and divides the parabola into two equal halves. The axis of symmetry of y = -1/2x2 is the y-axis. This is because the vertex is located on the y-axis, and the parabola is symmetric about the y-axis.
Intercepts
The intercepts of a parabola are the points where the parabola intersects the x-axis and y-axis. The x-intercepts of y = -1/2x2 are located at (0, 0) and (0, 0). The y-intercept of y = -1/2x2 is located at (0, 0).
Table of Values
The following table shows some of the key points on the graph of y = -1/2x2.
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1/2 |
| 0 | 0 |
| 1 | -1/2 |
| 2 | -2 |
How to Graph Y = 1/2x²
To graph the function y = 1/2x², follow these steps:
- Create a table of values by plugging in different values of x and solving for y.
- Plot the points from the table on the coordinate plane.
- Connect the points with a smooth curve to create the graph.
The graph of y = 1/2x² is a parabola that opens upward. The vertex of the parabola is at the origin (0, 0), and the axis of symmetry is the y-axis.
People Also Ask
How do I find the x-intercepts of y = 1/2x²?
To find the x-intercepts of y = 1/2x², set y = 0 and solve for x. This gives x = 0. Therefore, the only x-intercept is (0, 0).
How do I find the y-intercept of y = 1/2x²?
To find the y-intercept of y = 1/2x², set x = 0 and solve for y. This gives y = 0. Therefore, the y-intercept is (0, 0).
How do I find the vertex of y = 1/2x²?
The vertex of a parabola is the point where the parabola changes direction. The vertex of y = 1/2x² is at the origin (0, 0).
