Homework 5 Vertex Form Of A Quadratic Equation

Let's delve into the fascinating world of quadratic equations and explore the vertex form, a powerful tool for understanding and manipulating these mathematical expressions. This article will not only cover the basics but also equip you with the knowledge to confidently tackle homework problems involving the vertex form of a quadratic equation.

Unveiling the Quadratic Equation and Its Forms

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is expressed as:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The solutions to this equation, also known as roots or zeros, represent the x-intercepts of the parabola defined by the equation.

Besides the general form, quadratic equations can also be written in vertex form, which offers a unique perspective on the equation's properties. The vertex form is given by:

a(x - h)² + k = 0

Here, (h, k) represents the vertex of the parabola, the point where the parabola changes direction. The value of a remains the same as in the general form and determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), as well as the "width" of the parabola.

Why Vertex Form Matters: Advantages and Applications

The vertex form isn't just another way to write a quadratic equation; it offers several advantages that make it a valuable tool in various applications:

• Identifying the Vertex: The most apparent advantage is the immediate identification of the vertex (h, k). This is crucial for understanding the parabola's maximum or minimum point.

• Graphing Made Easy: Knowing the vertex and the direction the parabola opens allows for quick and accurate sketching of the graph.

• Solving Optimization Problems: Many real-world problems involve finding the maximum or minimum value of a quantity that can be modeled by a quadratic equation. The vertex form directly provides this optimal value. For example, determining the maximum height of a projectile or minimizing the cost of production.

• Understanding Transformations: The vertex form clearly shows how the basic parabola y = x² is transformed – horizontally by h units and vertically by k units.

Converting from General Form to Vertex Form: Completing the Square

The key to unlocking the power of the vertex form lies in the ability to convert a quadratic equation from its general form to vertex form. The most common method for this conversion is called "completing the square." Let's break down the process step-by-step:

1. Factor out 'a' from the x² and x terms:

Starting with the general form ax² + bx + c = 0, factor out the coefficient a from the first two terms:

a(x² + (b/a)x) + c = 0

2. Complete the square inside the parentheses:

To complete the square, take half of the coefficient of the x term (which is b/a), square it, and add it inside the parentheses. Remember to also subtract a times this value outside the parentheses to maintain the equation's balance.

Half of (b/a): (b/2a) Square of (b/2a): (b/2a)² = b²/4a²

Therefore, the equation becomes:

a(x² + (b/a)x + b²/4a²) + c - a(b²/4a²) = 0

3. Rewrite the expression inside the parentheses as a squared term:

The expression inside the parentheses is now a perfect square trinomial and can be rewritten as:

a(x + b/2a)² + c - b²/4a = 0

4. Simplify the constant term:

Combine the constant terms c and - b²/4a to get a single constant term:

a(x + b/2a)² + (4ac - b²)/4a = 0

5. Identify h and k:

Now the equation is in vertex form: a(x - h)² + k = 0. Comparing this with our transformed equation, we can identify h and k:

h = -b/2a k = (4ac - b²)/4a or k = c - b²/4a

Therefore, the vertex of the parabola is (-b/2a, (4ac - b²)/4a).

Example:

Convert the quadratic equation 2x² + 8x + 5 = 0 to vertex form.

1. Factor out 'a': 2(x² + 4x) + 5 = 0
2. Complete the square:
   • Half of 4: 2
   • Square of 2: 4
   • 2(x² + 4x + 4) + 5 - 2(4) = 0
3. Rewrite as a squared term: 2(x + 2)² + 5 - 8 = 0
4. Simplify the constant term: 2(x + 2)² - 3 = 0
5. Identify h and k: h = -2, k = -3

Therefore, the vertex form of the equation is 2(x + 2)² - 3 = 0, and the vertex of the parabola is (-2, -3).

Finding the Vertex Directly: Using the Formula

While completing the square is a fundamental technique, you can also find the vertex directly using a formula derived from the completing the square process. This is particularly useful when you only need the vertex coordinates and not the entire vertex form.

As we saw in the completing the square derivation, the x-coordinate of the vertex, h, is given by:

h = -b/2a

To find the y-coordinate of the vertex, k, substitute the value of h back into the original quadratic equation (in either general or vertex form):

k = f(h) = a(-b/2a)² + b(-b/2a) + c

or

k = a(h - h)² + k = k (using the vertex form)

Example:

Find the vertex of the quadratic equation x² - 6x + 8 = 0 using the formula.

1. Identify a, b, and c: a = 1, b = -6, c = 8
2. Calculate h: h = -(-6) / (2 * 1) = 3
3. Calculate k: k = (3)² - 6(3) + 8 = 9 - 18 + 8 = -1

Therefore, the vertex of the parabola is (3, -1).

Solving Quadratic Equations Using the Vertex Form

While the vertex form is excellent for understanding the parabola's properties, it can also be used to solve quadratic equations. This method is particularly useful when the equation is already in vertex form.

To solve a(x - h)² + k = 0 for x:

1. Isolate the squared term: a(x - h)² = -k
2. Divide by 'a': (x - h)² = -k/a
3. Take the square root of both sides: x - h = ±√(-k/a)
4. Solve for x: x = h ± √(-k/a)

Notice that if -k/a is negative, the solutions will be complex numbers. This corresponds to the case where the parabola does not intersect the x-axis.

Example:

Solve the quadratic equation 2(x - 1)² - 8 = 0 using the vertex form.

1. Isolate the squared term: 2(x - 1)² = 8
2. Divide by 'a': (x - 1)² = 4
3. Take the square root of both sides: x - 1 = ±√4 = ±2
4. Solve for x: x = 1 ± 2

Therefore, the solutions are x = 3 and x = -1.

Real-World Applications: Putting Vertex Form to Work

The vertex form of a quadratic equation finds its applications in various real-world scenarios. Here are a few examples:

• Projectile Motion: The height of a projectile (like a ball thrown in the air) as a function of time can often be modeled by a quadratic equation. The vertex represents the maximum height the projectile reaches. By expressing the equation in vertex form, you can easily determine this maximum height and the time it takes to reach it.

• Optimization Problems: Many business and engineering problems involve optimizing a quantity, such as maximizing profit or minimizing cost. If the quantity can be expressed as a quadratic function, the vertex form can be used to find the optimal value. For instance, determining the optimal production level to minimize average production cost.

• Architecture and Engineering: Parabolic arches are often used in bridges and buildings due to their structural properties. Understanding the vertex form of the quadratic equation describing the arch allows engineers to calculate the arch's height and width accurately.

• Curve Fitting: In data analysis, you might want to fit a quadratic curve to a set of data points. The vertex form can be useful in parameterizing the curve and understanding its characteristics.

Common Mistakes to Avoid

Working with the vertex form requires careful attention to detail. Here are some common mistakes to watch out for:

• Incorrectly Factoring out 'a': When completing the square, ensure you factor out 'a' correctly from both the x² and x terms.

• Forgetting to Adjust the Constant Term: When completing the square, remember to subtract a times the squared term that you added inside the parentheses from the constant term outside the parentheses to maintain the equation's balance.

• Misidentifying h and k: Remember that the vertex form is a(x - h)² + k = 0. Therefore, the x-coordinate of the vertex is h, not -h. Pay close attention to the sign.

• Incorrectly Applying the Formula: Double-check your calculations when using the formula h = -b/2a and when substituting h back into the equation to find k.

• Ignoring the Sign of 'a': The sign of 'a' determines whether the parabola opens upwards or downwards. This is crucial for interpreting the vertex as a maximum or minimum point.

Practice Problems: Sharpen Your Skills

To solidify your understanding of the vertex form, let's work through a few practice problems:

Problem 1:

Convert the quadratic equation y = -3x² + 12x - 5 to vertex form and identify the vertex.

Solution:

1. Factor out 'a': y = -3(x² - 4x) - 5
2. Complete the square:
   • Half of -4: -2
   • Square of -2: 4
   • y = -3(x² - 4x + 4) - 5 - (-3)(4)
3. Rewrite as a squared term: y = -3(x - 2)² - 5 + 12
4. Simplify the constant term: y = -3(x - 2)² + 7
5. Identify h and k: h = 2, k = 7

Therefore, the vertex form of the equation is y = -3(x - 2)² + 7, and the vertex is (2, 7).

Problem 2:

A ball is thrown vertically upwards with an initial velocity of 64 feet per second. Its height h (in feet) after t seconds is given by the equation h = -16t² + 64t. Find the maximum height reached by the ball.

Solution:

1. Find the vertex:
   • a = -16, b = 64, c = 0
   • t = -b/2a = -64 / (2 * -16) = 2
   • h = -16(2)² + 64(2) = -64 + 128 = 64

Therefore, the maximum height reached by the ball is 64 feet.

Problem 3:

Solve the quadratic equation * (x + 3)² - 4 = 0*.

Solution:

1. Isolate the squared term: (x + 3)² = 4
2. Take the square root of both sides: x + 3 = ±√4 = ±2
3. Solve for x: x = -3 ± 2

Therefore, the solutions are x = -1 and x = -5.

Beyond the Basics: Exploring Further Concepts

Once you've mastered the fundamentals of the vertex form, you can explore more advanced concepts, such as:

• Transformations of Quadratic Functions: Understanding how the parameters a, h, and k affect the graph of the parabola allows you to analyze and manipulate quadratic functions effectively.

• Applications in Calculus: The vertex form is useful in finding the maximum or minimum values of functions using calculus techniques, such as finding critical points.

• Quadratic Regression: Using statistical software to fit a quadratic model to data and interpret the resulting vertex form.

Conclusion: Mastering the Vertex Form

The vertex form of a quadratic equation is a powerful tool that provides valuable insights into the properties and behavior of parabolas. By mastering the techniques of converting between general form and vertex form, finding the vertex, and applying the vertex form to solve real-world problems, you'll gain a deeper understanding of quadratic equations and their applications in various fields. So, embrace the power of the vertex form and conquer your quadratic equation homework with confidence!