Axis Of Symmetry For Y=(x+5)(x-7) A Comprehensive Guide

When delving into the realm of quadratic equations, a fundamental concept to grasp is the axis of symmetry. This imaginary line carves the parabola, the graphical representation of a quadratic equation, into two perfectly symmetrical halves. Understanding how to pinpoint the axis of symmetry is crucial for analyzing and sketching quadratic functions accurately. In this comprehensive guide, we will dissect the equation y=(x+5)(x-7) and employ various methodologies to determine its axis of symmetry. We will explore the standard form of quadratic equations, the vertex form, and the method of averaging the roots, ensuring a thorough understanding of this vital concept.

Unveiling the Significance of the Axis of Symmetry

The axis of symmetry is not merely a mathematical construct; it's a cornerstone in understanding the behavior of quadratic functions. It provides valuable insights into the parabola's minimum or maximum point (the vertex), its direction of opening, and its overall shape. By accurately identifying the axis of symmetry, we gain a significant advantage in sketching the parabola and solving related problems, such as finding the vertex and determining the function's range. This knowledge is particularly beneficial in various real-world applications, from physics to engineering, where parabolic paths and shapes are frequently encountered.

Decoding the Quadratic Equation: Standard Form

To effectively find the axis of symmetry, it's essential to understand the standard form of a quadratic equation: y = ax² + bx + c. This form reveals key coefficients that play a pivotal role in determining the parabola's characteristics. The coefficient a dictates the parabola's direction of opening (upward if a is positive, downward if a is negative) and its vertical stretch or compression. The coefficients b and c influence the parabola's position on the coordinate plane. By converting the given equation, y=(x+5)(x-7), into standard form, we can readily identify these coefficients and utilize them to calculate the axis of symmetry.

Transforming y=(x+5)(x-7) into Standard Form

The first step in finding the axis of symmetry for the quadratic equation y=(x+5)(x-7) is to convert it into standard form, which is y = ax² + bx + c. To do this, we need to expand the factored form of the equation. This involves multiplying the two binomials, (x+5) and (x-7). Using the distributive property (often remembered by the acronym FOIL, which stands for First, Outer, Inner, Last), we multiply each term in the first binomial by each term in the second binomial:

• First: x * x = x²*
• Outer: x * (-7) = -7x*
• Inner: 5 * x = 5x
• Last: 5 * (-7) = -35

Now, we combine these terms: x² - 7x + 5x - 35. Next, we simplify by combining the like terms, which are the terms with x: -7x + 5x = -2x. So, the equation becomes y = x² - 2x - 35. Now the equation is in standard form, y = ax² + bx + c, where a = 1, b = -2, and c = -35. This standard form is crucial for various calculations related to quadratic equations, including finding the axis of symmetry and the vertex of the parabola.

Calculating the Axis of Symmetry using the Formula

Once the quadratic equation is in standard form, we can easily find the axis of symmetry using a simple formula. The axis of symmetry is a vertical line that passes through the vertex of the parabola, and its equation is given by x = -b / 2a, where a and b are the coefficients from the standard form of the quadratic equation, y = ax² + bx + c. In our equation, y = x² - 2x - 35, we identified that a = 1 and b = -2. Now, we can plug these values into the formula:

• x = -(-2) / (2 * 1)
• x = 2 / 2
• x = 1

Therefore, the axis of symmetry for the equation y = x² - 2x - 35 is the vertical line x = 1. This means that the parabola is symmetric about this line; if you were to fold the parabola along the line x = 1, the two halves would perfectly overlap. This axis of symmetry is a critical feature of the parabola, as it also helps to locate the vertex, which is the point where the parabola reaches its minimum or maximum value. In the next sections, we will further explore the significance of the axis of symmetry and how it relates to other properties of the quadratic equation.

Vertex Form: An Alternative Approach to Symmetry

Another powerful form for analyzing quadratic equations is the vertex form: y = a(x - h)² + k. In this form, the vertex of the parabola is readily apparent as the point (h, k). The h value represents the x-coordinate of the vertex, which also lies on the axis of symmetry. While converting to vertex form might seem like an additional step, it provides a direct pathway to identifying the axis of symmetry and the vertex, making it a valuable tool in our analytical arsenal.

Converting from Standard Form to Vertex Form

To convert the standard form of the quadratic equation, y = x² - 2x - 35, into vertex form, y = a(x - h)² + k, we need to complete the square. Completing the square is a technique that allows us to rewrite the quadratic expression as a perfect square trinomial plus a constant. This process involves several steps:

1. Isolate the x² and x terms: Start by isolating the terms containing x on one side of the equation. In our case, these terms are x² and -2x. So, we focus on the expression x² - 2x.
2. Complete the square: To complete the square, we need to add and subtract a value that will make the expression a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x - h)² or (x + h)². The value we need to add and subtract is (b / 2)², where b is the coefficient of the x term. In our equation, b = -2, so we calculate (-2 / 2)² = (-1)² = 1. We add and subtract this value within the expression: x² - 2x + 1 - 1.
3. Rewrite as a perfect square trinomial: Now, we can rewrite the first three terms as a perfect square trinomial. The expression x² - 2x + 1 can be factored into (x - 1)². So, we have (x - 1)² - 1.
4. Incorporate the constant term: Don't forget about the constant term from the original standard form, which is -35. We need to include this in our equation: y = (x - 1)² - 1 - 35.
5. Simplify: Finally, we simplify the equation by combining the constant terms: -1 - 35 = -36. So, the equation in vertex form is y = (x - 1)² - 36.

Now that the equation is in vertex form, y = a(x - h)² + k, we can easily identify the vertex of the parabola. In this case, h = 1 and k = -36, so the vertex is at the point (1, -36). The x-coordinate of the vertex, h, gives us the axis of symmetry, which is x = 1. This confirms the result we obtained earlier using the formula x = -b / 2a. The vertex form provides a clear and direct way to find the axis of symmetry and the vertex, which are essential for understanding the properties and graph of the quadratic equation.

Extracting the Axis of Symmetry from Vertex Form

Once the quadratic equation is in vertex form, y = a(x - h)² + k, identifying the axis of symmetry is straightforward. The axis of symmetry is a vertical line that passes through the vertex of the parabola. In the vertex form, the x-coordinate of the vertex is represented by h. Therefore, the equation of the axis of symmetry is simply x = h.

In our converted equation, y = (x - 1)² - 36, we can see that h = 1. This means the axis of symmetry is the vertical line x = 1. This confirms the axis of symmetry we found earlier using the formula x = -b / 2a and demonstrates the power of the vertex form in quickly determining key features of the parabola. The axis of symmetry is crucial because it divides the parabola into two symmetrical halves, making it easier to sketch and analyze the graph of the quadratic function. It also helps in locating the vertex, which is the maximum or minimum point of the parabola, and understanding the overall behavior of the function.

Averaging the Roots: A Root-Based Approach

A unique method for finding the axis of symmetry involves leveraging the roots (or x-intercepts) of the quadratic equation. The roots are the values of x that make y equal to zero. Due to the symmetry of the parabola, the axis of symmetry always lies exactly halfway between the two roots. Therefore, by simply averaging the roots, we can directly determine the axis of symmetry. This method is particularly effective when the quadratic equation is given in factored form, as the roots are readily apparent.

Determining the Roots from Factored Form

The given equation, y=(x+5)(x-7), is already in factored form, which makes it easy to identify the roots. The roots of a quadratic equation are the values of x that make y equal to zero. In factored form, the roots are the values of x that make each factor equal to zero. So, we set each factor equal to zero and solve for x:

1. Set the first factor equal to zero:
   • x + 5 = 0
   • Subtract 5 from both sides:
   • x = -5
2. Set the second factor equal to zero:
   • x - 7 = 0
   • Add 7 to both sides:
   • x = 7

Thus, the roots of the equation y=(x+5)(x-7) are x = -5 and x = 7. These roots represent the points where the parabola intersects the x-axis. Knowing the roots is essential for sketching the graph of the parabola and for finding the axis of symmetry, which, as we will see in the next step, is simply the average of the roots. This method provides a direct and intuitive way to find the axis of symmetry when the equation is given in factored form.

Calculating the Axis of Symmetry by Averaging the Roots

Now that we have identified the roots of the equation y = (x + 5)(x - 7) as x = -5 and x = 7, we can find the axis of symmetry by averaging these roots. The axis of symmetry is a vertical line that runs exactly in the middle of the two roots, due to the symmetrical nature of the parabola. To find the average of the roots, we add them together and divide by 2:

• Average of roots: (-5 + 7) / 2
• Simplify: (2) / 2
• Result: 1

Therefore, the axis of symmetry is the vertical line x = 1. This result confirms the axis of symmetry we found using the standard form formula (x = -b / 2a) and the vertex form method. Averaging the roots is a straightforward and efficient way to find the axis of symmetry, especially when the quadratic equation is given in factored form or when the roots are easily determined. This method highlights the connection between the roots and the symmetry of the parabola, providing a deeper understanding of quadratic functions.

The Answer

Through our exploration of the equation y=(x+5)(x-7), we've employed three distinct methods to pinpoint the axis of symmetry: utilizing the standard form formula, leveraging the vertex form, and averaging the roots. Each approach has consistently led us to the same conclusion: the axis of symmetry is located at x=1. This reinforces the fundamental symmetry inherent in quadratic equations and the versatility of these methods in solving related problems. Understanding these techniques empowers us to confidently analyze and sketch parabolas, unlocking a deeper understanding of quadratic functions.

Conclusion: Mastering the Axis of Symmetry

In conclusion, finding the axis of symmetry is a crucial skill in the study of quadratic equations. Whether we employ the standard form formula, delve into the vertex form, or average the roots, the axis of symmetry provides a vital reference point for understanding the parabola's behavior. By mastering these techniques, we gain a comprehensive understanding of quadratic functions and their applications in various fields. The ability to confidently determine the axis of symmetry not only enhances our problem-solving capabilities but also deepens our appreciation for the elegant symmetry inherent in mathematics.