Parabola Vertex, Focus, Directrix, And Axis Of Symmetry Explained

In this article, we will delve into the fascinating world of parabolas, exploring how to determine their key features such as the vertex, focus, directrix, and axis of symmetry. Understanding these elements is crucial for accurately sketching the graph of a parabola and comprehending its fundamental properties. We will tackle several examples, providing step-by-step solutions and clear explanations to guide you through the process. Whether you're a student learning about conic sections or simply seeking a refresher on parabolas, this comprehensive guide will equip you with the knowledge and skills to confidently analyze and graph these essential curves.

(a) x² = -4y

To begin, let's consider the equation x² = -4y. This equation represents a parabola that opens either upwards or downwards, as the x-term is squared. To identify the specific direction and other key features, we need to rewrite the equation in the standard form of a parabola with a vertical axis of symmetry. The standard form for such a parabola is (x - h)² = 4p(y - k), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus and from the vertex to the directrix.

Comparing x² = -4y with the standard form, we can see that h = 0, k = 0, and 4p = -4. This immediately tells us that the vertex of the parabola is at the origin, (0, 0). Solving 4p = -4 for p, we find that p = -1. Since p is negative, the parabola opens downwards. The focus of the parabola is located at (h, k + p), which in this case is (0, 0 + (-1)) = (0, -1). The directrix is a horizontal line given by the equation y = k - p, so here, it is y = 0 - (-1) = 1. Finally, the axis of symmetry is a vertical line passing through the vertex and the focus, which has the equation x = h, or x = 0. To sketch the graph, plot the vertex, focus, and directrix, and then draw a smooth curve that opens downwards, passing through the vertex and symmetric about the axis of symmetry. The focus is a crucial point because it defines the parabola's shape; all points on the parabola are equidistant from the focus and the directrix. The directrix acts as a boundary, ensuring that the parabola does not extend beyond a certain limit in the direction opposite the focus. Understanding the relationship between the vertex, focus, and directrix is fundamental to grasping the geometry of parabolas.

(b) 3y² = 24x

Now, let's analyze the equation 3y² = 24x. This equation represents a parabola that opens either to the left or to the right, as the y-term is squared. To determine the direction and other key features, we first need to rewrite the equation in the standard form of a parabola with a horizontal axis of symmetry. The standard form for such a parabola is (y - k)² = 4p(x - h), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus and from the vertex to the directrix.

First, divide both sides of the equation by 3 to get y² = 8x. Comparing this with the standard form, we can see that h = 0, k = 0, and 4p = 8. This indicates that the vertex of the parabola is at the origin, (0, 0). Solving 4p = 8 for p, we find that p = 2. Since p is positive, the parabola opens to the right. The focus of the parabola is located at (h + p, k), which in this case is (0 + 2, 0) = (2, 0). The directrix is a vertical line given by the equation x = h - p, so here, it is x = 0 - 2 = -2. The axis of symmetry is a horizontal line passing through the vertex and the focus, which has the equation y = k, or y = 0. To sketch the graph, plot the vertex, focus, and directrix, and then draw a smooth curve that opens to the right, passing through the vertex and symmetric about the axis of symmetry. The horizontal axis of symmetry means that the parabola is mirrored across the x-axis, and the positive value of p confirms that the parabola opens towards the positive x-axis. Accurately plotting these key points ensures that the sketched parabola accurately reflects the given equation.

(c) (y + 5/2)² = -5(x - 9/2)

Next, we consider the equation (y + 5/2)² = -5(x - 9/2). This equation is already in the standard form for a parabola with a horizontal axis of symmetry, which is (y - k)² = 4p(x - h). By comparing the given equation with the standard form, we can directly identify the key parameters. We have h = 9/2, k = -5/2, and 4p = -5. Thus, the vertex of the parabola is at (9/2, -5/2). Solving 4p = -5 for p, we find that p = -5/4. Since p is negative, the parabola opens to the left. The focus of the parabola is located at (h + p, k), which in this case is (9/2 + (-5/4), -5/2) = (13/4, -5/2). The directrix is a vertical line given by the equation x = h - p, so here, it is x = 9/2 - (-5/4) = 23/4. The axis of symmetry is a horizontal line passing through the vertex and the focus, which has the equation y = k, or y = -5/2. To sketch the graph, plot the vertex, focus, and directrix, and then draw a smooth curve that opens to the left, passing through the vertex and symmetric about the axis of symmetry. The negative value of p is crucial in determining that the parabola opens to the left, distinguishing it from parabolas that open to the right. The fractional coordinates of the vertex and focus demonstrate the importance of being comfortable working with fractions in the context of conic sections.

(d) x² + 6x + 8y = 7

Now, let's tackle the equation x² + 6x + 8y = 7. This equation is not in the standard form, so we need to complete the square to rewrite it in a recognizable format. Since the x-term is squared, this parabola will open either upwards or downwards. To complete the square for the x-terms, we take half of the coefficient of the x-term (which is 6), square it (which is 9), and add it to both sides of the equation. This gives us: x² + 6x + 9 + 8y = 7 + 9. We can now rewrite the left side as a perfect square: (x + 3)² + 8y = 16. Next, we isolate the squared term by subtracting 8y from both sides: (x + 3)² = -8y + 16. Factoring out -8 from the right side, we get (x + 3)² = -8(y - 2). Now the equation is in the standard form (x - h)² = 4p(y - k). Comparing this with our equation, we find that h = -3, k = 2, and 4p = -8. This tells us that the vertex of the parabola is at (-3, 2). Solving 4p = -8 for p, we get p = -2. Since p is negative, the parabola opens downwards. The focus of the parabola is located at (h, k + p), which in this case is (-3, 2 + (-2)) = (-3, 0). The directrix is a horizontal line given by the equation y = k - p, so here, it is y = 2 - (-2) = 4. The axis of symmetry is a vertical line passing through the vertex and the focus, which has the equation x = h, or x = -3. Sketching the graph involves plotting these key points and drawing a smooth curve opening downwards, symmetric about the line x = -3. Completing the square is a fundamental algebraic technique that allows us to transform general quadratic equations into standard forms, making it easier to identify key features of conic sections. The negative value of p again indicates a downward-opening parabola, and the shift in the vertex from the origin ((-3, 2)) highlights the importance of correctly applying the transformations represented by h and k.

(e) y² - 12x + 8y = -16

Finally, let's examine the equation y² - 12x + 8y = -16. Similar to the previous example, this equation is not in standard form, so we need to complete the square. Since the y-term is squared, this parabola will open either to the left or to the right. To complete the square for the y-terms, we take half of the coefficient of the y-term (which is 8), square it (which is 16), and add it to both sides of the equation. This gives us: y² + 8y + 16 - 12x = -16 + 16. We can now rewrite the left side as a perfect square: (y + 4)² - 12x = 0. Next, we isolate the squared term by adding 12x to both sides: (y + 4)² = 12x. Now the equation is in the standard form (y - k)² = 4p(x - h). Comparing this with our equation, we find that h = 0, k = -4, and 4p = 12. This tells us that the vertex of the parabola is at (0, -4). Solving 4p = 12 for p, we get p = 3. Since p is positive, the parabola opens to the right. The focus of the parabola is located at (h + p, k), which in this case is (0 + 3, -4) = (3, -4). The directrix is a vertical line given by the equation x = h - p, so here, it is x = 0 - 3 = -3. The axis of symmetry is a horizontal line passing through the vertex and the focus, which has the equation y = k, or y = -4. To sketch the graph, plot the vertex, focus, and directrix, and then draw a smooth curve that opens to the right, passing through the vertex and symmetric about the axis of symmetry. The positive value of p clearly indicates that this parabola opens to the right, and the vertex being off the x-axis ((0, -4)) emphasizes the shift from the standard form centered at the origin. The process of completing the square is a powerful tool for handling parabolas and other conic sections that are not initially presented in their standard forms.

In this comprehensive guide, we have systematically determined the vertex, focus, directrix, and axis of symmetry for various parabolas given their equations. By rewriting the equations in standard form, we were able to easily identify these key features and sketch the graphs of the parabolas. Understanding these concepts is fundamental for further exploration of conic sections and their applications in mathematics and other fields. Whether you are dealing with parabolas that open upwards, downwards, left, or right, the principles discussed here will provide a solid foundation for your analysis. The ability to manipulate equations into standard forms and extract relevant parameters is a crucial skill in analytical geometry and calculus.