Graphing quadratic equations is a fundamental skill in algebra. One common type of quadratic equation is y = 2(3x^2). It forms a U-shaped curve called a parabola. Imagine a roller coaster with a dip in the middle. The parabola of y = 2(3x^2) resembles this shape, with the vertex (or lowest point) located at the bottom of the dip. Understanding how to graph this equation is crucial for visualizing and analyzing quadratic functions.
The first step in graphing y = 2(3x^2) is to find the vertex. The x-coordinate of the vertex is given by the formula x = -b/(2a), where a and b are the coefficients of the quadratic equation. In this case, a = 3 and b = 0, so the x-coordinate of the vertex is 0. The y-coordinate of the vertex is then found by plugging this x-value back into the equation: y = 2(3(0)^2) = 0. Therefore, the vertex of the parabola is at the point (0, 0).
Once you have the vertex, you can plot additional points to help you sketch the parabola. To do this, choose different x-values, plug them into the equation, and calculate the corresponding y-values. For instance, if you choose x = 1, you get y = 2(3(1)^2) = 6. This gives you the point (1, 6). Similarly, if you choose x = -1, you get y = 2(3(-1)^2) = 6, giving you the point (-1, 6). Plotting these points and connecting them with a smooth curve will give you the graph of y = 2(3x^2).
Identifying the Key Elements of the Equation
To understand how to graph the equation y = 2 – 3x^2, it’s essential to identify its key elements: the slope, y-intercept, and vertex. These elements provide crucial information about the graph’s shape, position, and behavior.
Slope
The slope of a line represents the rate of change in the y-coordinate for a given change in the x-coordinate. In the equation y = 2 – 3x^2, the coefficient of x^2 is -3. This value represents the slope of the graph, which indicates that for every unit increase in x, the y-coordinate decreases by 3 units. Therefore, the slope of the parabola is -3.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis (x = 0). In the equation y = 2 – 3x^2, the constant term is 2. This value represents the y-intercept, indicating that the parabola intersects the y-axis at the point (0, 2). Therefore, the y-intercept is 2.
Vertex
The vertex of a parabola is the point at which it changes direction. It represents the minimum or maximum point of the parabola. The x-coordinate of the vertex can be calculated using the formula x = -b/2a, where a is the coefficient of x^2 (in this case, -3) and b is the coefficient of x (which is 0 in this equation). Therefore, the x-coordinate of the vertex is 0.
To find the y-coordinate of the vertex, substitute the x-coordinate back into the equation: y = 2 – 3(0)^2 = 2. Therefore, the vertex of the parabola is (0, 2).
| Key Element | Value |
|---|---|
| Slope | -3 |
| Y-Intercept | 2 |
| Vertex | (0, 2) |
Plotting the Vertex of the Parabola
The vertex of a parabola is the point where it changes direction. To find the vertex of the parabola y = 2 – 3x2, we need to first find the x-coordinate of the vertex.
The x-coordinate of the vertex is given by the formula , where a and b are the coefficients of the x2 and x terms, respectively.
In our case, a = -3 and b = 0, so the x-coordinate of the vertex is .
Now that we know the x-coordinate of the vertex, we can find the y-coordinate by substituting the x-coordinate back into the original equation.
Plugging into y = 2 – 3x2, we get .
Therefore, the vertex of the parabola y = 2 – 3x2 is the point (0, 2).
Determining the Axis of Symmetry
To graph the quadratic function y = 2 + 3x², we need to determine its axis of symmetry. The axis of symmetry for a parabola of the form y = ax² + bx + c is given by the equation x = -b/2a.
For the given function, y = 2 + 3x², we have a = 3 and b = 0. Substituting these values into the equation for the axis of symmetry, we get:
x = -b/2a = -(0)/(2 * 3) = 0
Therefore, the axis of symmetry for the graph of y = 2 + 3x² is x = 0. This means that the graph will be symmetric about the y-axis.
Finding Coordinates on the Axis of Symmetry
Once we have determined the axis of symmetry, we can find coordinates on the graph that lie on the axis. To do this, we substitute x = 0 into the equation of the function:
y = 2 + 3x²
y = 2 + 3(0)²
y = 2
Therefore, the point (0, 2) lies on the axis of symmetry.
Finding Intercepts with the Coordinate Axes
To find the intercepts of the graph with the coordinate axes, we can substitute the values of x and y into the equation and solve for the other variable.
x-intercept
To find the x-intercept, we set y = 0 and solve for x:
0 = 2x^2 + 3x
0 = x(2x + 3)
x = 0, -3/2
Thus, the x-intercepts are (0, 0) and (-3/2, 0).
y-intercept
To find the y-intercept, we set x = 0 and solve for y:
y = 2(0)^2 + 3(0)
y = 0
Thus, the y-intercept is (0, 0).
Plotting the Points
The table below summarizes the intercepts and corresponding points:
| Intercept | Point |
|---|---|
| x-intercept | (0, 0) |
| x-intercept | (-3/2, 0) |
| y-intercept | (0, 0) |
To plot the points, locate the corresponding coordinates on the coordinate plane and mark them with a dot. The graph of the equation will pass through these points.
Establishing the Direction of Opening
The coefficient of the squared term (x2) in the equation y = 2 – 3x2 is negative, which indicates that the parabola opens downward. This is because the negative sign reverses the direction of opening for parabolas.
Symmetrical Points
Since the quadratic equation y = 2 – 3x2 lacks a linear (x) term, the graph will be symmetrical with respect to the y-axis. That means for any given x value on the graph, there will be a corresponding point that is mirrored across the y-axis with the same y value but an opposite x value.
Vertex
The vertex of the parabola is the point where it changes direction. For a parabola that opens downward, like y = 2 – 3x2, the vertex is the highest point on the graph. To find the x-coordinate of the vertex, use the formula:
| x-coordinate of vertex: | x = -\frac{b}{2a} |
|---|
In this case, b = 0, so the x-coordinate of the vertex is x = 0. To find the y-coordinate, substitute this value back into the original equation:
y = 2 – 3(0)2
y = 2
Therefore, the vertex of the parabola is (0, 2).
Completing the Square to Rewrite the Equation
To complete the square, we need to add and subtract half of the coefficient of the x-term squared to the original equation. In this case, the coefficient of the x-term is 2, so half of that is 1.
We add and subtract 1² = 1 to the equation:
“`
y = 2x² – 3x + 1 – 1
“`
This gives us:
“`
y = 2(x² – 3/2x + 1/2) – 1 + 1
“`
Now we can factor the quadratic inside the parentheses:
“`
y = 2(x – 3/4)² – 1
“`
This is the equation of a parabola with vertex at (3/4, -1).
Table of values for the parabola y = 2(x – 3/4)² – 1
| x | y |
|---|---|
| 0 | -1 |
| 1/4 | -3/2 |
| 3/4 | -1 |
| 1 | 1/2 |
| 2 | 3 |
Sketching the Parabola Using the Equation
To graph the parabola y = 2 – 3x2, follow these steps:
1. Find the Vertex
The vertex is the point where the parabola changes direction. The x-coordinate of the vertex is -b/2a, where a and b are the coefficients of the x2 and x terms, respectively. In this case, a = -3 and b = 0, so the vertex is located at x = 0.
2. Find the Y-Intercept
The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, set x = 0 in the equation: y = 2 – 3(0)2 = 2. So, the y-intercept is (0, 2).
3. Find Additional Points
To get a better sense of the shape of the parabola, find a few additional points. Choose some values for x and solve for y:
| x | y |
|---|---|
| -1 | 5 |
| 1 | -1 |
| 2 | -10 |
4. Plot the Points
Plot the points (0, 2), (-1, 5), (1, -1), and (2, -10) on a graph.
5. Connect the Points
Connect the points with a smooth curve to form the parabola.
6. Determine the Symmetry
The parabola is symmetric about the x-axis because the equation is an even function.
7. Label the Graph
Clearly label the x- and y-axes, the vertex, and any other important points. Your final graph should look something like this:
[Image of a parabola with a vertex at (0, 2) and an opening downward]
Table of Contents
- Verifying the Graph Using a Calculator or Software
Verifying the Graph Using a Calculator or Software
Graphing calculators and software can simplify the task of graphing functions. They automate the creation of the graph, so you can confirm your hand-drawn graph or explore alternative perspectives.
Using a Graphing Calculator
Steps to verify the graph using a graphing calculator:
- Enter the function equation into the calculator.
- Adjust the viewing window to encompass the desired range.
- Graph the function and examine the resulting curve.
Using Graphing Software
Steps to verify the graph using graphing software:
- Input the function equation into the software interface.
- Set the appropriate graph parameters, such as axes limits and grid settings.
- Generate the graph and analyze its shape and characteristics.
Comparison of Methods
| Method | Advantages | Disadvantages |
|---|---|---|
| Hand-Drawing | Detailed and accurate, aids in understanding | Time-consuming, prone to errors |
| Calculator/Software | Quicker and convenient, automated | Limited precision, may not expose subtle features |
Both methods have their strengths and limitations. Hand-drawing provides a deeper understanding of the function’s behavior, while calculators or software expedite the process and reveal overall trends.
By utilizing a combination of methods, you can verify and enhance your understanding of the graph.
Analyzing the Function’s Behavior
To analyze the behavior of the function y = 2 – 3x2, it helps to calculate key values such as its vertex, axis of symmetry, and any intercepts.
Vertex
The vertex (h, k) is the point where the quadratic function attains its minimum or maximum value. For the given function, we have:
h = 0 (found by calculating the x-value of the vertex from h = –b/2a)
k = 2 (the y-value corresponding to h)
Therefore, the vertex of y = 2 – 3x2 is (0, 2).
Axis of Symmetry
The axis of symmetry is a vertical line passing through the vertex of the parabola. The equation of the axis of symmetry is:
x = h
For the given function, the axis of symmetry is x = 0.
y-Intercept
The y-intercept is the point where the graph crosses the y-axis. We can find it by setting x = 0 in the function:
y = 2 – 3(0)2
Therefore, the y-intercept is (0, 2).
x-Intercepts
The x-intercepts are the points where the graph crosses the x-axis. We can find them by setting y = 0 in the function:
0 = 2 – 3x2
Rearranging and solving for x, we get:
x = ±√(2/3)
Therefore, the x-intercepts are approximately (-0.82, 0) and (0.82, 0).
Interpreting the Graph in Context
The graph of y = 2 – 3x2 is a parabola that opens downward. The vertex of the parabola is the point at which the parabola changes direction. The x-coordinate of the vertex is the value of x at which the parabola reaches its maximum or minimum value. The y-coordinate of the vertex is the maximum or minimum value of the parabola.
The x-coordinate of the vertex is:
-1
The y-coordinate of the vertex is:
3
The vertex of the parabola is the point:
(-1, 3)
The axis of symmetry of the parabola is the vertical line that passes through the vertex. The equation of the axis of symmetry is:
x = -1
The graph of y = 2 – 3x2 is symmetric with respect to the axis of symmetry.
| X-intercept | Y-intercept |
|---|---|
| 1 | 2 |
The graph of y = 2 – 3x2 has two x-intercepts. The x-intercepts are the points at which the parabola intersects the x-axis. The x-intercepts are:
(1, 0) and (-1, 0)
The graph of y = 2 – 3x2 has one y-intercept. The y-intercept is the point at which the parabola intersects the y-axis. The y-intercept is:
(0, 2)
How to Graph Y = 2 – 3x2
To graph the equation y = 2 – 3x2, follow these steps:
- Find the vertex of the parabola by completing the square.
- Plot the vertex on the graph.
- Determine the direction of the parabola by looking at the sign of the coefficient of x2. If it is negative, the parabola opens downward. If it is positive, the parabola opens upward.
- Plot additional points on the parabola by plugging in values for x and solving for y.
- Connect the points to create the graph of the parabola.
People Also Ask
What is the vertex of the parabola y = 2 – 3x2?
The vertex of the parabola is at (0, 2).
Does the parabola y = 2 – 3x2 open upward or downward?
The parabola opens downward because the coefficient of x2 is negative.
