The Graph Of Which Function Will Have A Maximum And A $y$-intercept Of $4$?A. $f(x)=4x^2+6x-1$B. $f(x)=-4x^2+8x+5$C. $f(x)=-x^2+2x+4$D. $f(x)=x^2+4x-4$
Introduction
In mathematics, particularly in algebra and calculus, understanding the properties of functions is crucial. One of the essential properties of a function is its graph, which can be either a straight line or a curve. In this article, we will explore the characteristics of a function's graph, specifically focusing on the presence of a maximum and a y-intercept of 4.
What is a y-Intercept?
A y-intercept is the point at which a line or curve intersects the y-axis. In other words, it is the value of y when x is equal to 0. The y-intercept is an essential concept in mathematics, particularly in graphing functions.
What is a Maximum?
A maximum is the highest point on a curve or the highest value of a function. In the context of a function's graph, a maximum can occur at a single point or over a range of values.
Analyzing the Functions
To determine which function has a maximum and a y-intercept of 4, we need to analyze each option.
Option A: f(x) = 4x^2 + 6x - 1
This is a quadratic function, which means its graph is a parabola. To find the y-intercept, we substitute x = 0 into the function:
f(0) = 4(0)^2 + 6(0) - 1 f(0) = -1
This function does not have a y-intercept of 4.
Option B: f(x) = -4x^2 + 8x + 5
This is also a quadratic function. To find the y-intercept, we substitute x = 0 into the function:
f(0) = -4(0)^2 + 8(0) + 5 f(0) = 5
This function does not have a y-intercept of 4.
Option C: f(x) = -x^2 + 2x + 4
This is a quadratic function. To find the y-intercept, we substitute x = 0 into the function:
f(0) = -(0)^2 + 2(0) + 4 f(0) = 4
This function has a y-intercept of 4.
Option D: f(x) = x^2 + 4x - 4
This is a quadratic function. To find the y-intercept, we substitute x = 0 into the function:
f(0) = (0)^2 + 4(0) - 4 f(0) = -4
This function does not have a y-intercept of 4.
Conclusion
Based on the analysis of each option, we can conclude that the function f(x) = -x^2 + 2x + 4 has a maximum and a y-intercept of 4.
Why is this function special?
This function is special because it has a maximum at a single point, which is a characteristic of a quadratic function. The maximum point occurs when the derivative of the function is equal to 0.
Derivative of the Function
To find the derivative of the function f(x) = -x^2 + 2x + 4, we use the power rule of differentiation:
f'(x) = -2x + 2
Finding the Maximum Point
To find the maximum point, we set the derivative equal to 0 and solve for x:
-2x + 2 = 0 -2x = -2 x = 1
Conclusion
In conclusion, the function f(x) = -x^2 + 2x + 4 has a maximum and a y-intercept of 4. This function is special because it has a maximum at a single point, which is a characteristic of a quadratic function.
What is the significance of this function?
This function is significant because it represents a real-world scenario where a maximum value is achieved at a single point. For example, in economics, a company's profit may be represented by a function like f(x) = -x^2 + 2x + 4, where x is the number of units produced and the maximum profit is achieved at a single point.
Real-World Applications
This function has real-world applications in various fields, including economics, physics, and engineering. For example, in physics, the function f(x) = -x^2 + 2x + 4 may represent the motion of an object under the influence of a force, where the maximum velocity is achieved at a single point.
Conclusion
Introduction
In our previous article, we explored the characteristics of a function's graph, specifically focusing on the presence of a maximum and a y-intercept of 4. We analyzed each option and concluded that the function f(x) = -x^2 + 2x + 4 has a maximum and a y-intercept of 4. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the significance of a y-intercept?
A: A y-intercept is the point at which a line or curve intersects the y-axis. It is an essential concept in mathematics, particularly in graphing functions. The y-intercept represents the value of y when x is equal to 0.
Q: What is a maximum?
A: A maximum is the highest point on a curve or the highest value of a function. In the context of a function's graph, a maximum can occur at a single point or over a range of values.
Q: Why is the function f(x) = -x^2 + 2x + 4 special?
A: This function is special because it has a maximum at a single point, which is a characteristic of a quadratic function. The maximum point occurs when the derivative of the function is equal to 0.
Q: What is the derivative of the function f(x) = -x^2 + 2x + 4?
A: To find the derivative of the function f(x) = -x^2 + 2x + 4, we use the power rule of differentiation:
f'(x) = -2x + 2
Q: How do you find the maximum point of a function?
A: To find the maximum point, we set the derivative equal to 0 and solve for x:
-2x + 2 = 0 -2x = -2 x = 1
Q: What is the real-world application of the function f(x) = -x^2 + 2x + 4?
A: This function has real-world applications in various fields, including economics, physics, and engineering. For example, in physics, the function f(x) = -x^2 + 2x + 4 may represent the motion of an object under the influence of a force, where the maximum velocity is achieved at a single point.
Q: Can you provide more examples of functions with a maximum and a y-intercept of 4?
A: Yes, here are a few more examples of functions with a maximum and a y-intercept of 4:
- f(x) = -2x^2 + 4x + 4
- f(x) = x^2 - 2x + 4
- f(x) = -3x^2 + 6x + 4
Q: How do you determine if a function has a maximum and a y-intercept of 4?
A: To determine if a function has a maximum and a y-intercept of 4, you need to analyze the function and its derivative. You can use the following steps:
- Find the y-intercept by substituting x = 0 into the function.
- Find the derivative of the function using the power rule of differentiation.
- Set the derivative equal to 0 and solve for x to find the maximum point.
- Check if the y-intercept is equal to 4.
Conclusion
In conclusion, the function f(x) = -x^2 + 2x + 4 has a maximum and a y-intercept of 4. This function is special because it has a maximum at a single point, which is a characteristic of a quadratic function. The significance of this function lies in its real-world applications, where it can be used to model various scenarios, including economics, physics, and engineering.