Finding The Value Of B In Curve Equation Y(x) = X^2 - Ax + B

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In the realm of mathematics, quadratic curves hold a special place. These curves, defined by equations of the form y(x) = ax^2 + bx + c, are ubiquitous in various fields, from physics and engineering to economics and computer science. Understanding the properties of these curves, such as their turning points and intercepts, is crucial for solving a wide range of problems. In this comprehensive guide, we will delve into the intricacies of finding the value of 'b' in a quadratic curve given its turning point.

Understanding Quadratic Curves and Turning Points

Before we embark on our quest to find the value of 'b', let's first establish a solid understanding of quadratic curves and their turning points. A quadratic curve, also known as a parabola, is a symmetrical U-shaped curve. Its equation is characterized by the presence of a squared term (x^2), which gives the curve its distinctive shape. The turning point of a parabola is the point where the curve changes direction. This point can be either a minimum or a maximum, depending on whether the parabola opens upwards or downwards.

The Equation of a Quadratic Curve

The general equation of a quadratic curve is given by:

y(x) = ax^2 + bx + c

where 'a', 'b', and 'c' are constants. The coefficient 'a' determines the direction in which the parabola opens. If 'a' is positive, the parabola opens upwards, and the turning point is a minimum. If 'a' is negative, the parabola opens downwards, and the turning point is a maximum. The coefficients 'b' and 'c' affect the position and shape of the parabola.

Turning Points and Their Significance

The turning point of a parabola is a critical feature that provides valuable information about the curve. At the turning point, the slope of the curve is zero. This means that the derivative of the quadratic function at the turning point is equal to zero. This property is instrumental in finding the coordinates of the turning point.

The x-coordinate of the turning point can be found using the formula:

x_turning_point = -b / 2a

Once we have the x-coordinate, we can substitute it back into the original equation of the parabola to find the y-coordinate of the turning point.

The Problem at Hand

Now that we have a solid grasp of quadratic curves and turning points, let's turn our attention to the specific problem we aim to solve. We are given the curve:

y(x) = x^2 - ax + b

where 'a' and 'b' are constants. We are also given that the curve has a turning point at P(1, 3). Our goal is to find the value of 'b'.

Solution Strategy

To find the value of 'b', we will employ a two-pronged approach:

  1. Utilize the Turning Point Property: We know that the turning point occurs at x = 1. This means that the derivative of the function y(x) at x = 1 must be equal to zero. We will use this information to find the value of 'a'.
  2. Substitute the Turning Point Coordinates: We know that the turning point is at P(1, 3). This means that when x = 1, y(x) = 3. We will substitute these values into the equation of the curve and use the value of 'a' we found in the previous step to solve for 'b'.

Step 1: Finding the Value of 'a'

First, let's find the derivative of the function y(x):

dy/dx = 2x - a

Since the turning point occurs at x = 1, we set the derivative equal to zero and substitute x = 1:

2(1) - a = 0

Solving for 'a', we get:

a = 2

Step 2: Finding the Value of 'b'

Now that we have the value of 'a', we can substitute it back into the equation of the curve:

y(x) = x^2 - 2x + b

We also know that the turning point is at P(1, 3). This means that when x = 1, y(x) = 3. Let's substitute these values into the equation:

3 = (1)^2 - 2(1) + b

Simplifying the equation, we get:

3 = 1 - 2 + b

3 = -1 + b

Solving for 'b', we get:

b = 4

Therefore, the value of 'b' is 4.

Conclusion

In this comprehensive guide, we have successfully found the value of 'b' in the quadratic curve y(x) = x^2 - ax + b, given that it has a turning point at P(1, 3). We achieved this by leveraging the properties of quadratic curves and their turning points, specifically the fact that the derivative of the function at the turning point is zero. By applying this principle and substituting the coordinates of the turning point, we were able to solve for 'a' and subsequently for 'b'.

This exercise highlights the power of mathematical concepts in solving real-world problems. Understanding the properties of curves and their turning points is crucial for various applications, including optimization problems, trajectory analysis, and curve fitting.

By mastering these concepts, you will be well-equipped to tackle a wide range of mathematical challenges and gain a deeper appreciation for the beauty and elegance of mathematics.

Key Takeaways:

  • Quadratic curves are defined by equations of the form y(x) = ax^2 + bx + c.
  • The turning point of a parabola is the point where the curve changes direction.
  • The x-coordinate of the turning point can be found using the formula x = -b / 2a.
  • At the turning point, the derivative of the function is equal to zero.
  • By utilizing these properties, we can solve for unknown parameters in quadratic curves.

Further Exploration:

  • Explore the relationship between the coefficients of a quadratic equation and the shape and position of the parabola.
  • Investigate the applications of quadratic curves in various fields, such as physics, engineering, and economics.
  • Practice solving similar problems involving turning points and other properties of curves.

By delving deeper into these topics, you will enhance your understanding of mathematics and its applications in the world around us.