25. Which Of The Following Is The Maximum Value Of The Equation $y = -x^2 + 2x + 5$?A. 6 B. 1 C. 5 D. 2
Introduction
Quadratic equations are a fundamental concept in mathematics, and understanding how to maximize or minimize them is crucial in various fields, including physics, engineering, and economics. In this article, we will focus on maximizing the quadratic equation $y = -x^2 + 2x + 5$ and determine the maximum value of the equation.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c$, where a, b, and c are constants. The graph of a quadratic equation is a parabola, which can be either upward-facing (if a > 0) or downward-facing (if a < 0).
Maximizing Quadratic Equations
To maximize a quadratic equation, we need to find the vertex of the parabola. The vertex is the point on the parabola where the function reaches its maximum or minimum value. The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$.
Finding the Vertex
In our equation $y = -x^2 + 2x + 5$, we have a = -1 and b = 2. Plugging these values into the formula, we get:
So, the x-coordinate of the vertex is 1.
Finding the Maximum Value
To find the maximum value of the equation, we need to plug the x-coordinate of the vertex into the equation. Substituting x = 1 into the equation, we get:
Therefore, the maximum value of the equation $y = -x^2 + 2x + 5$ is 6.
Conclusion
In conclusion, maximizing quadratic equations is a crucial concept in mathematics, and understanding how to find the vertex and maximum value of a quadratic equation is essential. By following the steps outlined in this article, we can determine the maximum value of the equation $y = -x^2 + 2x + 5$, which is 6.
Comparison of Options
Let's compare our answer with the options provided:
- A. 6: This is the correct answer.
- B. 1: This is the x-coordinate of the vertex, not the maximum value.
- C. 5: This is the constant term in the equation, not the maximum value.
- D. 2: This is not the maximum value of the equation.
Final Answer
Introduction
In our previous article, we discussed how to maximize the quadratic equation $y = -x^2 + 2x + 5$ and determined the maximum value of the equation. In this article, we will provide a Q&A guide to help you better understand the concept of maximizing quadratic equations.
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c$, where a, b, and c are constants.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point on the parabola where the function reaches its maximum or minimum value. The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$.
Q: How do I find the vertex of a parabola?
A: To find the vertex of a parabola, you need to plug the values of a and b into the formula $x = -\frac{b}{2a}$. For example, in the equation $y = -x^2 + 2x + 5$, we have a = -1 and b = 2. Plugging these values into the formula, we get:
Q: How do I find the maximum value of a quadratic equation?
A: To find the maximum value of a quadratic equation, you need to plug the x-coordinate of the vertex into the equation. For example, in the equation $y = -x^2 + 2x + 5$, we found that the x-coordinate of the vertex is 1. Plugging x = 1 into the equation, we get:
Q: What is the difference between the x-coordinate of the vertex and the maximum value?
A: The x-coordinate of the vertex is the point on the parabola where the function reaches its maximum or minimum value. The maximum value is the actual value of the function at the x-coordinate of the vertex.
Q: Can I use the quadratic formula to find the maximum value of a quadratic equation?
A: No, the quadratic formula is used to find the roots of a quadratic equation, not the maximum value. To find the maximum value, you need to find the vertex of the parabola and plug the x-coordinate into the equation.
Q: What are some common mistakes to avoid when maximizing quadratic equations?
A: Some common mistakes to avoid when maximizing quadratic equations include:
- Not finding the vertex of the parabola
- Not plugging the x-coordinate of the vertex into the equation
- Using the quadratic formula to find the maximum value
- Not checking the sign of the coefficient of the x^2 term (a)
Conclusion
In conclusion, maximizing quadratic equations is a crucial concept in mathematics, and understanding how to find the vertex and maximum value of a quadratic equation is essential. By following the steps outlined in this article, you can better understand the concept of maximizing quadratic equations and avoid common mistakes.
Final Tips
- Make sure to find the vertex of the parabola before plugging the x-coordinate into the equation.
- Check the sign of the coefficient of the x^2 term (a) to determine whether the parabola is upward-facing or downward-facing.
- Use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.
- Plug the x-coordinate of the vertex into the equation to find the maximum value.