Write An Equation For The Parabola Graphed Below

Writing the Equation of a Parabola from its Graph

Understanding how to derive the equation of a parabola from its graph is a fundamental skill in algebra and pre-calculus. This seemingly simple task involves applying key concepts about parabolas, including their vertex, focus, directrix, and axis of symmetry. This article will guide you through the process, covering various scenarios and providing a deeper understanding of the underlying mathematics. We'll explore different forms of parabolic equations and demonstrate how to determine the correct form and parameters based on the information presented in the graph. By the end, you'll be confident in your ability to write the equation for any parabola presented graphically.

Understanding Parabolas: Key Properties

Before we dive into the equation-writing process, let's refresh our understanding of parabolas. A parabola is a U-shaped curve that is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

• Vertex: The vertex is the lowest or highest point on the parabola. It's the turning point of the curve.
• Focus: A fixed point inside the parabola. The distance from any point on the parabola to the focus is equal to its distance to the directrix.
• Directrix: A fixed straight line outside the parabola.
• Axis of Symmetry: A line that divides the parabola into two mirror-image halves. This line passes through the vertex and the focus.
• p: The distance between the vertex and the focus (or the vertex and the directrix). This value is crucial in determining the equation.

Standard Forms of Parabola Equations

Parabolas can open upwards, downwards, to the left, or to the right. Their equations are expressed in different standard forms depending on their orientation:

1. Parabola opening upwards or downwards:

The general equation is: y = a(x - h)² + k

Where:

• (h, k) are the coordinates of the vertex.
• a determines the parabola's width and direction. If a > 0, the parabola opens upwards; if a < 0, it opens downwards. The absolute value of a influences the parabola's width; a larger absolute value indicates a narrower parabola.

2. Parabola opening to the left or to the right:

The general equation is: x = a(y - k)² + h

Where:

• (h, k) are the coordinates of the vertex.
• a determines the parabola's width and direction. If a > 0, the parabola opens to the right; if a < 0, it opens to the left. The absolute value of a again influences the width.

Step-by-Step Guide: Writing the Equation

Let's walk through the process with a hypothetical example. Suppose we have a parabola opening upwards with a vertex at (2, 1) and passing through the point (4, 5).

Step 1: Identify the vertex and orientation.

From the graph, we determine the vertex (h, k) = (2, 1). Since the parabola opens upwards, we'll use the equation y = a(x - h)² + k.

Step 2: Substitute the vertex into the equation.

Substituting the vertex coordinates, we get: y = a(x - 2)² + 1.

Step 3: Use a known point to solve for 'a'.

The graph shows the parabola passes through (4, 5). We substitute these coordinates into the equation:

5 = a(4 - 2)² + 1

5 = 4a + 1

4a = 4

a = 1

Step 4: Write the final equation.

Now that we know a = 1, we can write the complete equation of the parabola:

y = (x - 2)² + 1

This equation accurately represents the parabola shown in the graph.

Working with Different Information: Focus and Directrix

Sometimes, instead of a second point, the graph might provide the focus and directrix. Let's consider a parabola opening to the right with vertex at (1, 2), focus at (3, 2), and directrix at x = -1.

Step 1: Determine the value of 'p'.

The distance between the vertex and the focus (or vertex and directrix) is 'p'. In this case, p = 2 (the distance between (1, 2) and (3, 2)).

Step 2: Determine the value of 'a'.

For parabolas opening to the right or left, the relationship between 'a' and 'p' is: a = 1/(4p). In our example:

a = 1/(4 * 2) = 1/8

Step 3: Use the vertex and 'a' to write the equation.

Since the parabola opens to the right, we use the equation x = a(y - k)² + h:

x = (1/8)(y - 2)² + 1

Handling Parabolas with a Horizontal Axis of Symmetry

If the parabola has a horizontal axis of symmetry, meaning it opens left or right, the process is similar but uses the equation x = a(y - k)² + h. You'll need either a second point or the focus and directrix to solve for 'a'.

For example, a parabola opening to the left with a vertex at (-1, 1) and passing through (-5, 3):

Step 1: Use the equation x = a(y - k)² + h.

Step 2: Substitute the vertex: x = a(y - 1)² - 1.

Step 3: Substitute the point (-5, 3): -5 = a(3 - 1)² - 1. This simplifies to -4 = 4a, so a = -1.

Step 4: The equation is x = -(y - 1)² - 1.

Dealing with Stretches and Compressions

The value of 'a' not only determines the direction of opening but also the stretch or compression of the parabola. A larger absolute value of 'a' makes the parabola narrower (vertical stretch), while a smaller absolute value makes it wider (vertical compression). The same principle applies to horizontal stretches and compressions when dealing with parabolas opening left or right.

Frequently Asked Questions (FAQ)

Q: What if the graph doesn't clearly show the vertex or a specific point?

A: You can still estimate the vertex from the graph. If accurate coordinates are unavailable, you can approximate them and accept a degree of error in your equation. Using more points to refine your 'a' value can help increase accuracy.

Q: Can I use different forms of the equation to represent the same parabola?

A: While the standard forms are most convenient, it's mathematically possible to manipulate the equation into other forms, but it's generally unnecessary and might complicate the process.

Q: What happens if the parabola is degenerate (a line or a point)?

A: A degenerate parabola wouldn't follow the standard equation forms we’ve discussed. It's a special case outside the scope of this explanation.

Q: How do I check if my equation is correct?

A: Substitute the coordinates of known points (vertex and at least one other point) from the graph into your equation. If the equation holds true for all points, then your equation is accurate. You can also graph your derived equation using graphing software or a calculator to visually compare it with the original graph.

Conclusion

Determining the equation of a parabola from its graph involves a systematic approach. By understanding the standard forms of parabolic equations, the significance of the vertex, focus, directrix, and the value of 'a' (and its relationship to 'p'), you can confidently derive accurate equations for various parabola orientations. Remember to carefully analyze the provided information and choose the appropriate equation form accordingly. Mastering this skill strengthens your understanding of conic sections and prepares you for more advanced mathematical concepts. Practice with various examples is key to building proficiency and confidence in this area. Remember to always check your work by substituting known points into your final equation.