Finding The Quadratic Function With A Maximum And Y-intercept Of 4

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Finding the right function whose graph exhibits specific characteristics, such as a maximum point and a defined y-intercept, can be an interesting mathematical puzzle. In this article, we will explore how to identify a quadratic function that meets these criteria. Specifically, we aim to find a function that has a maximum value and intersects the y-axis at the point (0, 4). This exploration involves understanding the properties of quadratic functions, including the significance of the leading coefficient and the constant term.

Understanding Quadratic Functions

To identify the function with a maximum and a y-intercept of 4, it's crucial to first understand the general form and properties of quadratic functions. A quadratic function is typically expressed in the form f(x) = ax² + 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 that opens either upwards or downwards depending on the sign of the coefficient a.

  • Coefficient a: This coefficient plays a critical role in determining the shape and orientation of the parabola. If a is positive (a > 0), the parabola opens upwards, indicating that the function has a minimum value. Conversely, if a is negative (a < 0), the parabola opens downwards, indicating that the function has a maximum value. Therefore, for a function to have a maximum, the coefficient a must be negative.
  • Coefficients b and c: While a determines the concavity of the parabola, b and c influence its position in the coordinate plane. The coefficient b is related to the axis of symmetry of the parabola, which is a vertical line that passes through the vertex (the maximum or minimum point) of the parabola. The c coefficient, on the other hand, directly represents the y-intercept of the function. The y-intercept is the point where the parabola intersects the y-axis, which occurs when x = 0. Thus, in the function f(x) = ax² + bx + c, the y-intercept is the point (0, c).
  • Vertex of the Parabola: The vertex is the point where the parabola changes direction. If the parabola opens upwards (a > 0), the vertex represents the minimum point of the function. If the parabola opens downwards (a < 0), the vertex represents the maximum point. The x-coordinate of the vertex can be found using the formula x = -b / (2a). Once the x-coordinate is known, the y-coordinate can be found by substituting this x-value back into the original function.

In summary, to identify a quadratic function with a maximum, we need to look for a function with a negative a coefficient. To ensure the y-intercept is 4, the constant term c must be equal to 4. This understanding forms the basis for analyzing the given options and selecting the correct function.

Analyzing the Given Functions

Now, let's analyze the given functions to determine which one has a maximum and a y-intercept of 4. We will examine each function based on the criteria established in the previous section: the sign of the leading coefficient (a) and the value of the constant term (c).

  1. f(x) = 4x² + 6x - 1

    • In this function, the coefficient a is 4, which is positive. This indicates that the parabola opens upwards, and thus, the function has a minimum value, not a maximum. Therefore, this function does not meet our criteria.
    • The constant term c is -1, which means the y-intercept is (0, -1). This also does not match our desired y-intercept of 4.

  2. f(x) = -4x² + 8x + 5

    • Here, the coefficient a is -4, which is negative. This indicates that the parabola opens downwards, meaning the function has a maximum value. This aligns with our first criterion.
    • The constant term c is 5, resulting in a y-intercept of (0, 5). While this function has a maximum, the y-intercept does not match our requirement of 4. Therefore, this function is also not the correct choice.

  3. f(x) = -x² + 2x + 4

    • In this function, the coefficient a is -1 (since -x² is the same as -1x²), which is negative. This means the parabola opens downwards, and the function has a maximum value. This meets our first criterion.
    • The constant term c is 4, which gives us a y-intercept of (0, 4). This perfectly matches our desired y-intercept. Therefore, this function satisfies both conditions: it has a maximum and a y-intercept of 4.

  4. f(x) = x² + 4x - 4

    • The coefficient a in this function is 1, which is positive. This indicates that the parabola opens upwards, and the function has a minimum value, not a maximum. Thus, this function does not meet our criteria.
    • The constant term c is -4, resulting in a y-intercept of (0, -4), which does not match our desired y-intercept of 4.

Through this analysis, we can clearly see that only one function meets both requirements: having a maximum and a y-intercept of 4. The process of elimination, based on the properties of quadratic functions, allows us to confidently identify the correct function.

The Correct Function

Based on the analysis, the function that has a maximum and a y-intercept of 4 is f(x) = -x² + 2x + 4. This conclusion is drawn from the two key properties we identified: the negative leading coefficient indicating a maximum, and the constant term of 4 ensuring the y-intercept is at (0, 4).

To further confirm this, let's delve deeper into the characteristics of this function:

  • Maximum Value: Since the coefficient a is -1, which is negative, the parabola opens downwards. This confirms that the function has a maximum value. To find the exact maximum value, we can determine the vertex of the parabola. The x-coordinate of the vertex is given by x = -b / (2a). In this case, a = -1 and b = 2, so x = -2 / (2 * -1) = 1. To find the y-coordinate of the vertex, we substitute x = 1 into the function: f(1) = -(1)² + 2(1) + 4 = -1 + 2 + 4 = 5. Therefore, the vertex of the parabola is at the point (1, 5), and the maximum value of the function is 5.
  • Y-intercept: The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when x = 0. Substituting x = 0 into the function f(x) = -x² + 2x + 4, we get f(0) = -(0)² + 2(0) + 4 = 4. This confirms that the y-intercept is indeed (0, 4), as required.

The graph of this function is a parabola that opens downwards, with its vertex at (1, 5) and crossing the y-axis at (0, 4). This visual representation further solidifies our conclusion that f(x) = -x² + 2x + 4 is the function that meets the specified criteria.

In conclusion, by understanding the properties of quadratic functions and analyzing the coefficients, we were able to successfully identify the function with a maximum and a y-intercept of 4. This process highlights the importance of understanding the relationship between the equation of a function and its graphical representation.

Significance of the y-intercept

The y-intercept holds significant importance in the context of functions and their graphs. It represents the point where the graph of the function intersects the y-axis, which is the vertical axis in the coordinate plane. In simpler terms, the y-intercept is the value of the function when the input x is equal to 0. This point is crucial for understanding the behavior and characteristics of a function, particularly in real-world applications and modeling scenarios.

  • Definition and Calculation: The y-intercept is formally defined as the point (0, y), where y is the value of the function when x is 0. To find the y-intercept of a function, you simply substitute x = 0 into the function's equation and solve for y. For example, in the quadratic function f(x) = ax² + bx + c, the y-intercept is found by setting x = 0, which gives f(0) = a(0)² + b(0) + c = c. Thus, the y-intercept is the point (0, c), and the value of y at this point is c.
  • Graphical Interpretation: Graphically, the y-intercept is the point where the graph of the function crosses the vertical y-axis. This point provides a clear visual representation of the function's value at x = 0. The y-intercept is a key feature to look for when sketching or analyzing the graph of a function, as it provides an immediate reference point on the coordinate plane.
  • Real-world Applications: The y-intercept has numerous practical applications across various fields. In many real-world scenarios, the y-intercept represents the initial value or starting point of a process or phenomenon being modeled by the function. For instance:
    • In financial models, the y-intercept might represent the initial investment or the starting balance of an account.
    • In physics, the y-intercept could represent the initial position of an object or the initial velocity of a moving body.
    • In business, the y-intercept might indicate the fixed costs of production, which are the costs incurred even when no units are produced.
    • In demographics, the y-intercept could represent the population size at the beginning of a study or observation period.
  • Significance in Linear Functions: In linear functions, which have the form f(x) = mx + b, the y-intercept is particularly straightforward to interpret. Here, b represents the y-intercept, and it signifies the value of y when x is 0. The y-intercept, in combination with the slope m, completely defines the linear function and its position on the coordinate plane.
  • Significance in Quadratic Functions: In quadratic functions, the y-intercept is the constant term c in the standard form f(x) = ax² + bx + c. As we saw in the main problem, the y-intercept is a crucial characteristic that helps in identifying and analyzing quadratic functions. It provides one of the key anchor points for sketching the parabola and understanding its vertical position.
  • In Polynomial and Other Functions: The concept of the y-intercept extends to polynomial functions and other types of functions as well. In any function, the y-intercept is found by evaluating the function at x = 0. This point is invaluable for understanding the function's behavior and for graphical representation.

In summary, the y-intercept is a fundamental concept in the study of functions, offering both a graphical and practical understanding of the function's initial state or value. Its significance spans various disciplines and plays a crucial role in modeling and interpreting real-world phenomena.

Conclusion

In the realm of quadratic functions, identifying specific characteristics such as a maximum value and a y-intercept requires a solid understanding of the function's properties and coefficients. By systematically analyzing the given functions and applying the principles of quadratic equations, we successfully pinpointed the function f(x) = -x² + 2x + 4 as the one that possesses a maximum and a y-intercept of 4.

This exploration underscores the importance of understanding the role of coefficients in determining the shape and position of a parabola. The negative leading coefficient indicated a downward-opening parabola with a maximum value, while the constant term directly revealed the y-intercept. These insights not only help in solving specific problems but also enhance our overall comprehension of quadratic functions and their graphical representations.

The process of analyzing and identifying functions based on their properties is a fundamental skill in mathematics. It allows us to connect algebraic expressions with graphical representations, providing a more complete and intuitive understanding of mathematical concepts. By mastering these skills, we can confidently tackle more complex problems and applications in mathematics and related fields. This journey through quadratic functions and their characteristics highlights the beauty and practicality of mathematical analysis.