Analyzing G(t) = T^2 - T - 42 A Comprehensive Guide To Finding Roots And Vertex

Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically the function g(t) = t^2 - t - 42. We'll explore how to find its roots (where the function crosses the x-axis) and determine its vertex (the highest or lowest point on the parabola). So, buckle up and let's get started!

Understanding Quadratic Functions

Before we jump into the specifics of g(t), let's refresh our understanding of quadratic functions in general. These functions are defined by the standard form: f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve. The coefficient 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). This is a key concept to remember, as it influences the function's overall behavior and the nature of its vertex.

In our case, g(t) = t^2 - t - 42 is a quadratic function where a = 1, b = -1, and c = -42. Since 'a' is positive (1), we know that the parabola opens upwards, meaning it has a minimum point – the vertex. Understanding this fundamental property helps us visualize the function's graph and anticipate the characteristics of its solutions. Moreover, the constants 'b' and 'c' play crucial roles in determining the position and shape of the parabola. 'b' influences the axis of symmetry, while 'c' represents the y-intercept, providing valuable anchors for graphing and analysis. Analyzing these coefficients gives us a comprehensive understanding of the function's behavior. Quadratic functions are ubiquitous in various fields, from physics (projectile motion) to engineering (designing parabolic reflectors) and economics (modeling cost curves). Their widespread applicability highlights the importance of mastering their properties and analysis techniques. By understanding the core principles of quadratic functions, we can effectively tackle a wide range of problems and gain a deeper appreciation for their role in mathematical modeling.

Finding the Roots of g(t) = t^2 - t - 42

The roots of a function are the values of 't' for which g(t) = 0. In simpler terms, they are the points where the parabola intersects the t-axis. There are a couple of ways we can find these roots:

Factoring the Quadratic

This method involves breaking down the quadratic expression into two binomial factors. We're looking for two numbers that multiply to -42 (the 'c' term) and add up to -1 (the 'b' term). After a little thought, we can see that -7 and 6 fit the bill. So, we can rewrite the function as:

g(t) = (t - 7)(t + 6)

To find the roots, we set each factor equal to zero:

t - 7 = 0 => t = 7 t + 6 = 0 => t = -6

Therefore, the roots of g(t) are t = 7 and t = -6. Factoring is a powerful technique, but it's not always straightforward, especially when the coefficients are large or the roots are not integers. This method hinges on the ability to recognize the correct factors, which can sometimes be challenging. However, when factoring is possible, it provides a direct and efficient way to find the roots. The roots represent the points where the parabola intersects the t-axis, giving us crucial information about the function's behavior. These points are also known as the zeros of the function, as they are the values of t that make g(t) equal to zero. Understanding the significance of the roots is essential for analyzing and interpreting quadratic functions in various contexts. Furthermore, the factored form of the quadratic equation provides valuable insights into the function's behavior and allows us to quickly determine the roots without resorting to more complex methods like the quadratic formula. This is a fundamental skill in algebra and calculus.

Using the Quadratic Formula

If factoring doesn't work (or if you're not a fan of it!), the quadratic formula is your trusty backup. It's a foolproof method for finding the roots of any quadratic equation. The formula is:

t = [-b ± √(b^2 - 4ac)] / 2a

Plugging in our values (a = 1, b = -1, c = -42), we get:

t = [1 ± √((-1)^2 - 4 * 1 * -42)] / 2 * 1 t = [1 ± √(1 + 168)] / 2 t = [1 ± √169] / 2 t = [1 ± 13] / 2

So, we have two solutions:

t = (1 + 13) / 2 = 7 t = (1 - 13) / 2 = -6

Again, we find that the roots are t = 7 and t = -6. The quadratic formula is a cornerstone of algebra, providing a universal solution for finding the roots of any quadratic equation. Its reliability stems from its derivation based on completing the square, a method that transforms the quadratic equation into a perfect square trinomial. The formula's power lies in its ability to handle any quadratic, regardless of the nature of its coefficients or the roots themselves. The discriminant (b^2 - 4ac) within the formula plays a crucial role, as its value determines the number and type of roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. This information is invaluable for understanding the function's behavior and the characteristics of its graph. The quadratic formula is not just a mathematical tool; it's a testament to the elegance and power of algebraic manipulation, providing a reliable solution for a fundamental problem in mathematics and its applications.

Determining the Vertex of g(t) = t^2 - t - 42

The vertex is the turning point of the parabola. Since our parabola opens upwards (a > 0), the vertex represents the minimum point of the function. There are a couple of ways to find it:

Using the Vertex Formula

The x-coordinate (in our case, the t-coordinate) of the vertex can be found using the formula:

t_vertex = -b / 2a

For g(t), this gives us:

t_vertex = -(-1) / 2 * 1 = 1 / 2

To find the y-coordinate (g(t_vertex)), we plug this value back into the function:

g(1/2) = (1/2)^2 - (1/2) - 42 g(1/2) = 1/4 - 1/2 - 42 g(1/2) = -1/4 - 42 g(1/2) = -169/4

So, the vertex is at (1/2, -169/4). The vertex formula is a direct and efficient method for finding the turning point of a parabola, providing crucial information about the function's minimum or maximum value. It's derived from the process of completing the square, which transforms the quadratic equation into vertex form, directly revealing the coordinates of the vertex. The formula highlights the relationship between the coefficients of the quadratic equation and the position of the vertex. The t-coordinate of the vertex (-b/2a) represents the axis of symmetry of the parabola, a vertical line that divides the parabola into two symmetrical halves. Understanding this symmetry is invaluable for graphing and analyzing quadratic functions. The g(t) value of the vertex represents the minimum or maximum value of the function, depending on whether the parabola opens upwards (a > 0) or downwards (a < 0). This information is crucial for optimization problems, where we seek to find the minimum or maximum value of a function under certain constraints. The vertex formula is a fundamental tool in the study of quadratic functions, providing a clear and concise way to determine the key features of the parabola and its associated function. Its simplicity and effectiveness make it an indispensable technique for anyone working with quadratic equations.

Completing the Square

Another way to find the vertex is by completing the square. This involves rewriting the quadratic function in vertex form: g(t) = a(t - h)^2 + k, where (h, k) is the vertex. Let's do it:

g(t) = t^2 - t - 42 g(t) = (t^2 - t) - 42

To complete the square, we need to add and subtract (b/2a)^2 inside the parentheses. In this case, (b/2a)^2 = (-1/2)^2 = 1/4:

g(t) = (t^2 - t + 1/4 - 1/4) - 42 g(t) = (t - 1/2)^2 - 1/4 - 42 g(t) = (t - 1/2)^2 - 169/4

Now we can clearly see that the vertex is at (1/2, -169/4). Completing the square is a powerful algebraic technique that not only helps find the vertex of a parabola but also provides a deeper understanding of the structure of quadratic functions. It involves transforming the quadratic expression into a perfect square trinomial, which can then be easily factored. This method reveals the vertex form of the quadratic equation, g(t) = a(t - h)^2 + k, where (h, k) represents the vertex. The process of completing the square involves adding and subtracting a specific constant to the quadratic expression, ensuring that the overall value of the expression remains unchanged. This constant is calculated as (b/2a)^2, where 'a' and 'b' are the coefficients of the quadratic equation. Completing the square is not just a method for finding the vertex; it's also a fundamental technique for solving quadratic equations, especially when factoring is not straightforward. By transforming the equation into a form where one side is a perfect square, we can easily solve for the variable by taking the square root. Moreover, completing the square provides insights into the symmetry and transformations of quadratic functions, helping us understand how changes in the coefficients affect the graph of the parabola. It's a versatile and valuable skill in algebra, providing a pathway to solve a wide range of problems involving quadratic expressions and equations.

Conclusion

So, there you have it! We've successfully analyzed the quadratic function g(t) = t^2 - t - 42, finding its roots (t = 7 and t = -6) and its vertex (1/2, -169/4). We explored two methods for finding both the roots (factoring and the quadratic formula) and the vertex (the vertex formula and completing the square). Understanding these techniques is crucial for working with quadratic functions and solving related problems. Guys, keep practicing, and you'll become quadratic function masters in no time! Remember, math is like a muscle – the more you exercise it, the stronger it gets! We've covered a lot of ground today, from understanding the basic properties of quadratic functions to applying specific techniques for finding their roots and vertices. These skills are essential for further studies in mathematics, particularly in calculus and advanced algebra. The ability to analyze quadratic functions is not just about solving equations; it's about understanding the behavior of parabolas and their applications in various fields, from physics to engineering. Keep exploring, keep questioning, and keep building your mathematical foundation. The world of mathematics is vast and fascinating, and quadratic functions are just one piece of the puzzle. By mastering these fundamentals, you're setting yourself up for success in future mathematical endeavors. So, go forth and conquer those quadratic functions!