Which Equation Represents The Parabola Shown On The Graph

Which Equation Represents the Parabola Shown on the Graph? A Comprehensive Guide

Determining the equation of a parabola from its graph involves understanding the parabola's key features and applying the appropriate formula. This guide will walk you through the process step-by-step, covering various forms of parabolic equations and how to identify the correct one based on visual cues from the graph. We'll explore the vertex form, standard form, and intercept form, emphasizing the importance of correctly identifying the vertex, focus, directrix, and intercepts.

Understanding the Anatomy of a Parabola

Before diving into equations, let's refresh our understanding of a parabola's essential elements:

• Vertex: The highest or lowest point on the parabola. It represents the parabola's turning point.
• Focus: A fixed point inside the parabola. All points on the parabola are equidistant from the focus and the directrix.
• Directrix: A fixed line outside the parabola. All points on the parabola are equidistant from the focus and the directrix.
• Axis of Symmetry: A line that divides the parabola into two symmetrical halves. It passes through the vertex and the focus.
• Intercepts: The points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept).

Parabola Equations: Different Forms, Different Approaches

Parabolas can be represented by several equations, each offering a different perspective and highlighting specific characteristics:

1. Vertex Form: Revealing the Vertex at a Glance

The vertex form of a parabola's equation is:

y = a(x - h)² + k

where:

• (h, k) represents 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' affects the parabola's width – a larger absolute value means a narrower parabola.

How to use it: If the graph clearly shows the vertex, this form is the most straightforward. Simply substitute the vertex coordinates (h, k) into the equation. Then, use another point on the parabola to solve for 'a'.

Example: Let's say the vertex is (2, 3) and the parabola passes through the point (4, 7).

1. Substitute the vertex into the equation: y = a(x - 2)² + 3
2. Substitute the point (4, 7) into the equation: 7 = a(4 - 2)² + 3
3. Solve for 'a': 4 = 4a => a = 1
4. The equation is: y = (x - 2)² + 3

2. Standard Form: Useful for Finding Intercepts and the Vertex

The standard form of a parabolic equation is:

y = ax² + bx + c

where:

• 'a', 'b', and 'c' are constants. 'a' determines the direction and width, similar to the vertex form.

How to use it: This form is beneficial when the y-intercept and other points are readily available from the graph. You can substitute the coordinates of three points from the graph to create a system of three equations with three unknowns (a, b, and c). Solving this system yields the values of a, b, and c, completing the equation.

Example: Suppose the parabola passes through points (0, 1), (1, 3), and (2, 7).

1. Use the points to create three equations:

   • 1 = a(0)² + b(0) + c => c = 1
   • 3 = a(1)² + b(1) + c => a + b + 1 = 3
   • 7 = a(2)² + b(2) + c => 4a + 2b + 1 = 7

2. Solve the system of equations: From the second equation, a + b = 2. From the third equation, 4a + 2b = 6, which simplifies to 2a + b = 3.

3. Subtracting the simplified second equation from the simplified third equation gives a = 1. Substituting a = 1 into a + b = 2 gives b = 1.

4. Therefore, the equation is: y = x² + x + 1

The x-coordinate of the vertex in the standard form is given by x = -b / 2a. Substitute this value into the equation to find the y-coordinate of the vertex.

3. Intercept Form: Ideal when x-intercepts are known

The intercept form (also called factored form) is:

y = a(x - p)(x - q)

where:

• 'p' and 'q' are the x-intercepts (the points where the parabola crosses the x-axis).

How to use it: This form is particularly helpful if the graph clearly displays the x-intercepts. Substitute the x-intercepts into the equation. Then, use another point on the parabola to solve for 'a'.

Example: Let's assume the x-intercepts are -1 and 3, and the parabola passes through (1, -4).

1. Substitute the x-intercepts: y = a(x + 1)(x - 3)
2. Substitute the point (1, -4): -4 = a(1 + 1)(1 - 3)
3. Solve for 'a': -4 = -4a => a = 1
4. The equation is: y = (x + 1)(x - 3)

Identifying the Correct Equation: A Step-by-Step Approach

1. Identify the Vertex: Locate the vertex of the parabola on the graph. This is crucial for using the vertex form.

2. Determine the Direction: Does the parabola open upwards (a > 0) or downwards (a < 0)?

3. Find the Intercepts: Note the x-intercepts and the y-intercept. The x-intercepts are useful for the intercept form, and the y-intercept is helpful for the standard form.

4. Choose the Appropriate Form: Based on the information gathered (vertex, intercepts, direction), select the most convenient form of the parabolic equation.

5. Substitute and Solve: Substitute the known values (vertex coordinates, intercepts, and at least one other point) into the chosen equation and solve for the unknown constant(s).

6. Verify: Check if the obtained equation accurately represents the parabola shown on the graph by plugging in a few additional points from the graph.

Advanced Considerations: Focus and Directrix

For a more precise determination of the parabolic equation, especially when dealing with less clear graphs, you might need to consider the focus and directrix. The general equation of a parabola with a vertical axis of symmetry is:

(x - h)² = 4p(y - k)

where:

• (h, k) is the vertex.
• 'p' is the distance between the vertex and the focus (and also between the vertex and the directrix). If the parabola opens upwards, p > 0; if it opens downwards, p < 0.

Similarly, for a parabola with a horizontal axis of symmetry:

(y - k)² = 4p(x - h)

Using the focus and directrix allows for a more rigorous determination of the equation, especially when the parabola's intercepts aren't clearly defined on the graph. The distance from any point on the parabola to the focus must equal the distance from that point to the directrix. This property provides an alternative approach to finding the equation.

Conclusion: Mastering Parabola Equations

Determining the equation of a parabola shown on a graph is a crucial skill in algebra and calculus. By understanding the different forms of parabolic equations and their relationship to the parabola's features (vertex, focus, directrix, intercepts), you can confidently analyze graphs and derive the correct equation. Remember to carefully analyze the graph, choose the appropriate equation form, and verify your result. Practice is key to mastering this process. Work through numerous examples, and soon you'll be able to effortlessly identify the equation representing any parabola.