Vertex And X-Intercepts Of Y=x²-4x-21: A Step-by-Step Guide

by Alex Johnson 60 views

\nUnderstanding quadratic equations is a cornerstone of algebra, and finding the vertex and x-intercepts is crucial for graphing and analyzing these equations. In this article, we'll break down the process of finding the vertex and x-intercepts of the quadratic equation y = x² - 4x - 21. We will explore different methods and provide clear, step-by-step instructions to help you master this essential skill. Whether you're a student tackling homework or just brushing up on your math, this guide will provide you with the knowledge and confidence to solve similar problems. Let's dive in and demystify quadratic equations together!

Understanding Quadratic Equations

Before we jump into solving the equation y = x² - 4x - 21, let's take a moment to understand quadratic equations in general. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x) is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic equation is a parabola, a U-shaped curve. The key features of a parabola that we're interested in are the vertex and the x-intercepts.

  • Vertex: The vertex is the point where the parabola changes direction. It's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). The vertex is a critical point for understanding the behavior of the quadratic function. To find the vertex, we often use the vertex form of a quadratic equation, or we can use a formula derived from completing the square. The x-coordinate of the vertex can be found using the formula x = -b / 2a, and the y-coordinate can be found by substituting this x-value back into the original equation. Knowing the vertex helps us determine the range and symmetry of the parabola.
  • X-intercepts: The x-intercepts are the points where the parabola intersects the x-axis. These are also known as the roots or zeros of the equation. At these points, the value of y is 0. Finding x-intercepts is essential for solving many real-world problems, such as determining when a projectile hits the ground. The x-intercepts can be found by setting the quadratic equation equal to zero and solving for x. This can be done through factoring, completing the square, or using the quadratic formula. Each method provides a different approach, and choosing the most efficient one depends on the specific equation. Understanding the x-intercepts helps us visualize the solutions of the quadratic equation and their significance. The discriminant (b² - 4ac) can also tell us how many real x-intercepts exist: two if it's positive, one if it's zero, and none if it's negative.

Finding the Vertex of y = x² - 4x - 21

Now, let's find the vertex of the given equation: y = x² - 4x - 21. We'll use the formula for the x-coordinate of the vertex:

x = -b / 2a

In our equation, a = 1 and b = -4. Plugging these values into the formula, we get:

x = -(-4) / (2 * 1) = 4 / 2 = 2

So, the x-coordinate of the vertex is 2. To find the y-coordinate, we substitute x = 2 back into the original equation:

y = (2)² - 4(2) - 21 = 4 - 8 - 21 = -25

Therefore, the vertex of the parabola is (2, -25). This point represents the minimum value of the function since the coefficient of the x² term (a) is positive, indicating the parabola opens upwards. The vertex is a crucial reference point for graphing the parabola and understanding its behavior. Knowing the vertex allows us to easily identify the axis of symmetry, which is a vertical line passing through the vertex. In this case, the axis of symmetry is x = 2. Understanding the vertex not only helps in graphing but also in solving optimization problems where we need to find the minimum or maximum value of a quadratic function.

Finding the X-Intercepts of y = x² - 4x - 21

Next, we need to find the x-intercepts of the equation y = x² - 4x - 21. The x-intercepts are the points where the graph crosses the x-axis, meaning y = 0. So, we need to solve the equation:

x² - 4x - 21 = 0

We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to -21 and add up to -4. Those numbers are -7 and 3. So, we can factor the equation as:

(x - 7)(x + 3) = 0

Now, we set each factor equal to zero and solve for x:

x - 7 = 0 => x = 7 x + 3 = 0 => x = -3

Thus, the x-intercepts are (7, 0) and (-3, 0). These points indicate where the parabola intersects the x-axis. The x-intercepts are also the solutions or roots of the quadratic equation. In practical terms, they can represent the points where a projectile lands on the ground or the values at which a business breaks even. The factored form of the quadratic equation makes it easy to identify these intercepts, providing a direct way to visualize the function's behavior. Knowing the x-intercepts, along with the vertex, gives us a comprehensive picture of the parabola's shape and position on the coordinate plane. This information is invaluable for graphing the function accurately and solving related problems.

Putting It All Together

So, for the equation y = x² - 4x - 21, we have found:

  • Vertex: (2, -25)
  • X-intercepts: (7, 0) and (-3, 0)

These key points give us a good understanding of the parabola's shape and position. The vertex (2, -25) tells us the lowest point of the parabola, and the x-intercepts (7, 0) and (-3, 0) show where it crosses the x-axis. With this information, we can accurately graph the parabola and analyze its behavior. Understanding these components is essential for various applications, including physics, engineering, and economics, where quadratic functions often model real-world phenomena. By knowing the vertex and intercepts, we can determine the maximum or minimum values, the points of equilibrium, and other critical aspects of the system being modeled. This comprehensive approach to analyzing quadratic equations provides a solid foundation for more advanced mathematical concepts and problem-solving.

Graphing the Parabola

To graph the parabola y = x² - 4x - 21, we can use the information we've already found:

  1. Plot the vertex at (2, -25).
  2. Plot the x-intercepts at (7, 0) and (-3, 0).

Since parabolas are symmetrical, we can also find additional points to help us sketch the graph. For example, we can find the y-intercept by setting x = 0 in the original equation:

y = (0)² - 4(0) - 21 = -21

So, the y-intercept is (0, -21). Plotting this point gives us a better sense of the parabola's shape. Now, we can draw a smooth U-shaped curve through these points, ensuring the parabola is symmetrical around the vertical line passing through the vertex (the axis of symmetry). Graphing the parabola provides a visual representation of the quadratic function, making it easier to understand its behavior and characteristics. The graph allows us to see the relationship between the x and y values, the minimum value of the function (at the vertex), and the points where the function equals zero (the x-intercepts). This visual tool is invaluable for solving problems and making predictions based on the quadratic model. By combining the algebraic analysis with the graphical representation, we gain a complete understanding of the quadratic function.

Conclusion

In this article, we've walked through the process of finding the vertex and x-intercepts of the quadratic equation y = x² - 4x - 21. We found that the vertex is (2, -25) and the x-intercepts are (7, 0) and (-3, 0). These points are essential for understanding the behavior and graphing of the parabola represented by this equation. Mastering these skills is crucial for success in algebra and beyond.

Understanding quadratic equations opens doors to solving various real-world problems in fields like physics, engineering, and economics. The vertex helps us find maximum or minimum values, while x-intercepts show us when the function equals zero. By learning these fundamental concepts, you build a strong foundation for future mathematical studies and practical applications. Keep practicing and exploring different quadratic equations to solidify your understanding. Remember, each equation is a new puzzle, and finding its vertex and intercepts is the key to solving it. Happy problem-solving!

For further learning and practice, you might find helpful resources on websites like Khan Academy's Quadratic Equations Section.