Graphing H(x) = -1/3x^2 + 2x - 4 A Step-by-Step Guide
In mathematics, understanding how to graph functions is a fundamental skill, especially when dealing with quadratic functions. Quadratic functions, characterized by the general form f(x) = ax^2 + bx + c, create a parabola when graphed. This article will delve into a comprehensive guide on graphing the quadratic function h(x) = -1/3x^2 + 2x - 4. We'll explore various methods, including finding the vertex, intercepts, and axis of symmetry, to accurately sketch the graph. Mastering these techniques not only enhances your mathematical abilities but also provides a solid foundation for more advanced concepts in calculus and other related fields.
Understanding Quadratic Functions
Quadratic functions are polynomial functions of the second degree, meaning the highest power of the variable x is 2. The general form of a quadratic function is given by f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. If a > 0, the parabola opens upwards, indicating a minimum value, while if a < 0, the parabola opens downwards, indicating a maximum value.
In the given function, h(x) = -1/3x^2 + 2x - 4, we can identify the coefficients as a = -1/3, b = 2, and c = -4. Since a is negative, the parabola will open downwards, meaning it has a maximum point. The vertex of the parabola is this maximum (or minimum) point, and it is a critical feature to determine when graphing the function. The coefficient a also affects the width of the parabola; a smaller absolute value of a results in a wider parabola, while a larger absolute value makes it narrower.
The term bx affects the position of the parabola’s axis of symmetry, which is a vertical line passing through the vertex that divides the parabola into two symmetrical halves. The constant term c represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. Understanding these key components of a quadratic function is essential for accurately graphing the function and interpreting its behavior. We will use these elements to graph h(x) effectively.
Key Steps to Graphing h(x) = -1/3x^2 + 2x - 4
Graphing the quadratic function h(x) = -1/3x^2 + 2x - 4 involves a series of methodical steps to ensure accuracy. These steps include finding the vertex, determining the axis of symmetry, locating the intercepts (both x and y), and plotting additional points to complete the graph. By breaking down the process into manageable steps, we can effectively visualize the parabola and understand its characteristics.
1. Finding the Vertex
The vertex of a parabola is the point where the function reaches its maximum or minimum value. For a quadratic function in the form f(x) = ax^2 + bx + c, the x-coordinate of the vertex, often denoted as h, can be found using the formula:
h = -b / 2a
In our function, h(x) = -1/3x^2 + 2x - 4, we have a = -1/3 and b = 2. Plugging these values into the formula, we get:
h = -2 / (2 * (-1/3)) = -2 / (-2/3) = 3
So, the x-coordinate of the vertex is 3. To find the y-coordinate, often denoted as k, we substitute h = 3 back into the function:
k = h(3) = -1/3(3)^2 + 2(3) - 4 = -1/3(9) + 6 - 4 = -3 + 6 - 4 = -1
Thus, the vertex of the parabola is the point (3, -1). This point is crucial because it serves as the central point around which the parabola is symmetrical.
2. Determining the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. The equation of the axis of symmetry is given by:
x = h
Where h is the x-coordinate of the vertex. In our case, since the x-coordinate of the vertex is 3, the axis of symmetry is the vertical line:
x = 3
This line is essential for graphing the parabola because it provides a reference for the symmetrical distribution of points. Any point on one side of the axis of symmetry has a corresponding point on the other side at the same vertical distance from the x-axis. This symmetry helps in plotting the parabola accurately.
3. Locating the Intercepts
Intercepts are the points where the parabola intersects the coordinate axes. There are two types of intercepts: the y-intercept and the x-intercept(s).
Y-intercept
The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function:
h(0) = -1/3(0)^2 + 2(0) - 4 = -4
So, the y-intercept is the point (0, -4). This point is straightforward to find and provides a quick reference for the parabola’s vertical position.
X-intercepts
The x-intercepts are the points where the parabola intersects the x-axis. These occur when h(x) = 0. To find the x-intercepts, we need to solve the quadratic equation:
-1/3x^2 + 2x - 4 = 0
To simplify, we can multiply the entire equation by -3 to eliminate the fraction:
x^2 - 6x + 12 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 1, b = -6, and c = 12. Plugging these values into the quadratic formula, we get:
x = (6 ± √((-6)^2 - 4(1)(12))) / 2(1) x = (6 ± √(36 - 48)) / 2 x = (6 ± √(-12)) / 2
Since the discriminant (b^2 - 4ac) is negative (-12), there are no real solutions for x. This means the parabola does not intersect the x-axis. Consequently, there are no x-intercepts.
4. Plotting Additional Points
To get a more accurate graph, it's helpful to plot additional points on either side of the vertex. Since the parabola is symmetrical about the axis of symmetry x = 3, we can choose x-values that are equidistant from the axis of symmetry. For example, we can choose x = 1 and x = 5.
For x = 1:
h(1) = -1/3(1)^2 + 2(1) - 4 = -1/3 + 2 - 4 = -1/3 - 2 = -7/3 ≈ -2.33
So, we have the point (1, -7/3).
For x = 5:
h(5) = -1/3(5)^2 + 2(5) - 4 = -1/3(25) + 10 - 4 = -25/3 + 6 = -25/3 + 18/3 = -7/3 ≈ -2.33
So, we have the point (5, -7/3). As expected, the y-values for x = 1 and x = 5 are the same due to the symmetry of the parabola.
Similarly, we can choose x = 2 and x = 4:
For x = 2:
h(2) = -1/3(2)^2 + 2(2) - 4 = -1/3(4) + 4 - 4 = -4/3 ≈ -1.33
So, we have the point (2, -4/3).
For x = 4:
h(4) = -1/3(4)^2 + 2(4) - 4 = -1/3(16) + 8 - 4 = -16/3 + 4 = -16/3 + 12/3 = -4/3 ≈ -1.33
So, we have the point (4, -4/3).
Graphing the Parabola
Now that we have all the key points and information, we can sketch the graph of h(x) = -1/3x^2 + 2x - 4.
1. Plot the Vertex
First, plot the vertex at (3, -1). This is the highest point on the parabola since the parabola opens downwards.
2. Draw the Axis of Symmetry
Draw a vertical dashed line through x = 3. This line serves as a guide for the symmetry of the parabola.
3. Plot the Y-intercept
Plot the y-intercept at (0, -4).
4. Plot Additional Points
Plot the additional points we calculated: (1, -7/3), (5, -7/3), (2, -4/3), and (4, -4/3). These points help to define the shape of the parabola.
5. Sketch the Parabola
Finally, draw a smooth curve through the plotted points, ensuring it is symmetrical about the axis of symmetry. The parabola should open downwards, with the vertex as the highest point.
Characteristics of the Graph
By examining the graph, we can observe several characteristics of the function h(x) = -1/3x^2 + 2x - 4:
- The parabola opens downwards because the coefficient of x^2 is negative (a = -1/3).
- The vertex (3, -1) represents the maximum point of the function.
- The axis of symmetry is the line x = 3.
- The y-intercept is (0, -4).
- There are no x-intercepts, as the parabola does not intersect the x-axis.
- The parabola is wider compared to y = x^2 due to the fractional value of a.
Graphing the quadratic function h(x) = -1/3x^2 + 2x - 4 involves a systematic approach that includes finding the vertex, axis of symmetry, intercepts, and additional points. By following these steps, we can accurately sketch the parabola and understand its key features. This comprehensive guide provides a strong foundation for graphing quadratic functions and applying these concepts in various mathematical contexts. Mastering these skills not only enhances your understanding of functions but also prepares you for more advanced mathematical studies. Remember to practice these steps with different quadratic functions to reinforce your knowledge and improve your graphing abilities.
