How To Find Y Coordinate Of Vertex

Finding the y-coordinate of the vertex of a parabola is a fundamental skill in algebra, with applications in various fields ranging from physics to economics. The vertex represents the highest or lowest point on the parabola, making its y-coordinate crucial for determining the maximum or minimum value of a quadratic function.

Understanding the Vertex of a Parabola

A parabola is a symmetrical U-shaped curve defined by a quadratic equation, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants. The vertex of this parabola is the point where the curve changes direction. If a > 0, the parabola opens upwards, and the vertex is the minimum point. Conversely, if a < 0, the parabola opens downwards, and the vertex is the maximum point.

The vertex of a parabola can be expressed as a coordinate point (h, k), where h represents the x-coordinate and k represents the y-coordinate. The x-coordinate, h, can be found using the formula h = -b / 2a. Once you have the x-coordinate, finding the y-coordinate, k, involves substituting h back into the original quadratic equation.

Methods to Find the Y-Coordinate of the Vertex

There are several methods to find the y-coordinate of the vertex, each with its own advantages depending on the given information. Here are the most common approaches:

1. Using the Vertex Formula

The most direct method is to use the vertex formula. This involves two steps:

• Step 1: Find the x-coordinate (h) Use the formula h = -b / 2a to find the x-coordinate of the vertex. This formula is derived from completing the square and represents the axis of symmetry of the parabola.
• Step 2: Find the y-coordinate (k) Substitute the value of h you found in Step 1 back into the original quadratic equation f(x) = ax² + bx + c. This gives you k = f(h) = a(h)² + b(h) + c, which is the y-coordinate of the vertex.

Example:

Consider the quadratic equation f(x) = 2x² - 8x + 6.

• a = 2, b = -8, and c = 6
• h = -(-8) / (2 * 2) = 8 / 4 = 2
• k = f(2) = 2(2)² - 8(2) + 6 = 8 - 16 + 6 = -2

Therefore, the vertex is (2, -2), and the y-coordinate of the vertex is -2.

2. Completing the Square

Completing the square is a method used to rewrite the quadratic equation in vertex form, which directly reveals the coordinates of the vertex.

• Step 1: Rewrite the equation Start with the standard form of the quadratic equation f(x) = ax² + bx + c.
• Step 2: Factor out 'a' from the x² and x terms Rewrite the equation as f(x) = a(x² + (b/a)x) + c.
• Step 3: Complete the square Take half of the coefficient of the x term inside the parentheses (b/2a), square it ((b/2a)² = b²/4a²), and add and subtract it inside the parentheses. This doesn't change the equation's value but allows us to complete the square. The equation becomes f(x) = a(x² + (b/a)x + b²/4a² - b²/4a²) + c.
• Step 4: Rewrite as a perfect square Rewrite the expression inside the parentheses as a perfect square: f(x) = a((x + b/2a)² - b²/4a²) + c.
• Step 5: Distribute 'a' and simplify Distribute the a and combine the constant terms: f(x) = a(x + b/2a)² - b²/4a + c. Often written as f(x) = a(x - h)² + k, where h = -b/2a and k = c - b²/4a.

The equation is now in vertex form f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. The y-coordinate of the vertex is k.

Example:

Using the same equation f(x) = 2x² - 8x + 6:

• f(x) = 2(x² - 4x) + 6
• Take half of -4, which is -2, and square it: (-2)² = 4
• Add and subtract 4 inside the parentheses: f(x) = 2(x² - 4x + 4 - 4) + 6
• Rewrite as a perfect square: f(x) = 2((x - 2)² - 4) + 6
• Distribute and simplify: f(x) = 2(x - 2)² - 8 + 6 = 2(x - 2)² - 2

The equation is now in vertex form f(x) = 2(x - 2)² - 2. The vertex is (2, -2), and the y-coordinate of the vertex is -2.

3. Using Calculus (Finding Critical Points)

If you are familiar with calculus, you can find the vertex by finding the critical points of the quadratic function.

• Step 1: Find the first derivative Take the first derivative of the quadratic function f(x) = ax² + bx + c. The derivative is f'(x) = 2ax + b.
• Step 2: Set the derivative equal to zero To find the critical points, set f'(x) = 0 and solve for x. This gives you 2ax + b = 0, so x = -b / 2a. This is the x-coordinate of the vertex, h.
• Step 3: Find the y-coordinate Substitute the value of h back into the original function f(x) to find the y-coordinate, k. k = f(h) = a(h)² + b(h) + c.

Example:

Using the equation f(x) = 2x² - 8x + 6:

• f'(x) = 4x - 8
• Set f'(x) = 0: 4x - 8 = 0 => 4x = 8 => x = 2
• k = f(2) = 2(2)² - 8(2) + 6 = 8 - 16 + 6 = -2

The vertex is (2, -2), and the y-coordinate of the vertex is -2.

4. Averaging the Roots (x-intercepts)

If the quadratic equation has real roots (x-intercepts), you can find the x-coordinate of the vertex by averaging the roots. The axis of symmetry passes through the vertex and is located exactly in the middle of the two roots.

• Step 1: Find the roots Solve the quadratic equation ax² + bx + c = 0 for x. You can use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This will give you two roots, x₁ and x₂.
• Step 2: Find the x-coordinate of the vertex Average the roots to find the x-coordinate of the vertex: h = (x₁ + x₂) / 2.
• Step 3: Find the y-coordinate Substitute the value of h back into the original function f(x) to find the y-coordinate, k. k = f(h) = a(h)² + b(h) + c.

Example:

Consider the equation f(x) = x² - 5x + 6.

• Find the roots: x = (5 ± √((-5)² - 4 * 1 * 6)) / (2 * 1) = (5 ± √1) / 2 => x₁ = 3, x₂ = 2
• Find the x-coordinate of the vertex: h = (3 + 2) / 2 = 2.5
• Find the y-coordinate: k = f(2.5) = (2.5)² - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25

The vertex is (2.5, -0.25), and the y-coordinate of the vertex is -0.25.

5. Using Symmetry

The axis of symmetry of a parabola always passes through the vertex. If you know another point on the parabola and its x-coordinate, and you also know the x-coordinate of the vertex, you can use symmetry to find the y-coordinate of the vertex.

• Step 1: Understand Symmetry Recognize that for every point (x, y) on the parabola, there is a corresponding point (2h - x, y) that is equidistant from the axis of symmetry (x = h).
• Step 2: Find a corresponding point If you know a point (x₁, y₁) on the parabola and the x-coordinate of the vertex h, you can find the y-coordinate of the vertex k.
• Step 3: Use the properties of quadratic functions While this method doesn’t directly calculate k, understanding the symmetrical nature helps in visualizing or verifying the vertex. Other methods are generally more efficient for direct calculation.

This method is more conceptual and less computational, primarily used to verify results obtained through other methods.

Practical Applications

Finding the y-coordinate of the vertex has numerous practical applications:

• Optimization Problems: In business, it can be used to determine the production level that maximizes profit or minimizes cost.
• Physics: In projectile motion, the y-coordinate of the vertex represents the maximum height reached by a projectile.
• Engineering: In designing parabolic reflectors (e.g., satellite dishes or solar concentrators), knowing the vertex is crucial for focusing energy or signals.
• Mathematics: It is fundamental in understanding the behavior and properties of quadratic functions and parabolas.

Tips and Tricks

• Double-Check Your Work: Carefully review your calculations, especially when dealing with negative signs.
• Use a Calculator: When dealing with complex numbers, a calculator can help reduce errors.
• Sketch the Parabola: A quick sketch can help you visualize the vertex and ensure that your answer makes sense.
• Understand the Significance: Always remember what the y-coordinate of the vertex represents in the context of the problem. Is it a maximum or a minimum value?

Common Mistakes to Avoid

• Incorrectly Applying the Formula: Ensure you correctly identify a, b, and c and substitute them into the correct places in the formula.
• Arithmetic Errors: Be careful with arithmetic, especially when squaring numbers or dealing with negative signs.
• Forgetting to Substitute: After finding the x-coordinate, remember to substitute it back into the original equation to find the y-coordinate.
• Misinterpreting the Results: Ensure you understand whether the vertex represents a maximum or minimum value based on the sign of a.

Conclusion

Finding the y-coordinate of the vertex of a parabola is an essential skill in algebra with wide-ranging applications. Whether you choose to use the vertex formula, complete the square, use calculus, or average the roots, understanding the underlying principles and practicing regularly will help you master this skill. By avoiding common mistakes and double-checking your work, you can confidently solve problems involving quadratic functions and parabolas. The y-coordinate of the vertex provides valuable information about the maximum or minimum value of the function, making it a crucial concept in various fields.