Finding The Quadratic Function With A Vertex At (2,-9)

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In the realm of quadratic functions, the vertex holds a position of paramount importance. It represents the minimum or maximum point on the parabola, dictating the overall shape and behavior of the function. When faced with the task of identifying a quadratic function with a specific vertex, a comprehensive understanding of vertex form and its implications becomes essential. This article delves into the process of determining the quadratic function with a vertex at (2, -9), exploring the underlying concepts and demonstrating the step-by-step approach to arrive at the solution.

Understanding Vertex Form

The vertex form of a quadratic function provides a direct and insightful representation of the parabola's vertex. It is expressed as:

f(x) = a(x - h)^2 + k

where:

  • (h, k) represents the coordinates of the vertex.
  • a determines the direction of opening and the stretch or compression of the parabola.

By carefully analyzing the vertex form, we can readily extract the vertex coordinates and gain valuable insights into the quadratic function's characteristics. The value of 'h' indicates the horizontal shift of the parabola from the origin, while 'k' represents the vertical shift. The coefficient 'a' dictates whether the parabola opens upwards (a > 0) or downwards (a < 0), and its magnitude influences the parabola's width.

Identifying the Function with Vertex (2, -9)

Our objective is to pinpoint the quadratic function that possesses a vertex at (2, -9). To achieve this, we will leverage the vertex form and systematically analyze the provided options. We are given four quadratic functions:

  1. f(x) = -(x - 1)(x - 5)
  2. f(x) = (x + 8)^2
  3. f(x) = -(x - 3)^2
  4. f(x) = (x - 5)(x + 1)

To determine the function with the desired vertex, we will transform each option into vertex form and compare the resulting vertex coordinates with (2, -9).

Option 1: f(x) = -(x - 1)(x - 5)

To convert this function into vertex form, we first expand the expression:

f(x) = -(x^2 - 6x + 5)
f(x) = -x^2 + 6x - 5

Next, we complete the square to rewrite the function in vertex form:

f(x) = -(x^2 - 6x) - 5
f(x) = -(x^2 - 6x + 9) - 5 + 9
f(x) = -(x - 3)^2 + 4

The vertex form of this function is f(x) = -(x - 3)^2 + 4, indicating a vertex at (3, 4). This does not match our target vertex of (2, -9), so option 1 is not the correct answer.

Option 2: f(x) = (x + 8)^2

This function is already in a form that is close to vertex form. We can rewrite it as:

f(x) = (x + 8)^2 + 0

This represents a parabola with a vertex at (-8, 0). This also does not match our target vertex of (2, -9), so option 2 is incorrect.

Option 3: f(x) = -(x - 3)^2

This function is also close to vertex form. We can rewrite it as:

f(x) = -(x - 3)^2 + 0

This indicates a vertex at (3, 0), which again does not align with our desired vertex of (2, -9). Therefore, option 3 is not the solution.

Option 4: f(x) = (x - 5)(x + 1)

Similar to option 1, we need to expand and complete the square to transform this function into vertex form:

f(x) = x^2 - 4x - 5

Completing the square:

f(x) = (x^2 - 4x) - 5
f(x) = (x^2 - 4x + 4) - 5 - 4
f(x) = (x - 2)^2 - 9

This function, f(x) = (x - 2)^2 - 9, is in vertex form and clearly shows a vertex at (2, -9). This matches our target vertex, making option 4 the correct answer.

Conclusion

Through a systematic analysis of the given quadratic functions and a thorough understanding of vertex form, we have successfully identified the function with a vertex at (2, -9). By converting each function into vertex form and comparing the resulting vertex coordinates, we determined that f(x) = (x - 2)^2 - 9 is the quadratic function that satisfies the specified condition. This exercise highlights the significance of vertex form in understanding and manipulating quadratic functions.

In the realm of mathematics, quadratic functions hold a prominent position, serving as fundamental building blocks in various fields, including physics, engineering, and economics. These functions, characterized by their parabolic graphs, exhibit a unique set of properties that make them invaluable tools for modeling real-world phenomena. Among these properties, the vertex stands out as a particularly significant feature, dictating the parabola's extreme point and influencing its overall behavior. In this comprehensive exploration, we delve deeper into the intricacies of quadratic functions and their vertices, unraveling their mathematical essence and highlighting their practical applications.

The General Form of a Quadratic Function

A quadratic function is formally defined as a polynomial function of degree two, expressed in the general form:

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

where 'a', 'b', and 'c' are constants, and 'a' is non-zero. The coefficient 'a' plays a crucial role in determining the parabola's orientation: a positive 'a' signifies an upward-opening parabola, while a negative 'a' indicates a downward-opening parabola. The coefficients 'b' and 'c' influence the parabola's position and shape within the coordinate plane.

Unveiling the Vertex: The Parabola's Extremum

The vertex of a parabola represents its extreme point, either the minimum or maximum value of the quadratic function. For an upward-opening parabola (a > 0), the vertex corresponds to the minimum point, while for a downward-opening parabola (a < 0), the vertex represents the maximum point. The vertex coordinates, denoted as (h, k), provide a concise summary of the parabola's location and extreme value.

Determining the Vertex Coordinates

There are several methods to determine the vertex coordinates of a quadratic function. One common approach involves using the vertex formula, which directly calculates the x-coordinate (h) of the vertex:

h = -b / 2a

Once the x-coordinate is obtained, the y-coordinate (k) can be found by substituting 'h' back into the original quadratic function:

k = f(h) = a(-b / 2a)^2 + b(-b / 2a) + c

Another method involves completing the square, which transforms the quadratic function into vertex form, as discussed earlier. The vertex coordinates can then be directly read from the vertex form equation.

The Significance of the Vertex

The vertex holds immense significance in understanding and interpreting quadratic functions. Its position provides crucial information about the parabola's symmetry and extreme behavior. The x-coordinate of the vertex represents the axis of symmetry, a vertical line that divides the parabola into two mirror-image halves. The y-coordinate of the vertex signifies the minimum or maximum value of the function, depending on the parabola's orientation.

Applications of Quadratic Functions and Vertices

Quadratic functions and their vertices find extensive applications in various fields:

  • Physics: Modeling projectile motion, where the vertex represents the maximum height reached by the projectile.
  • Engineering: Designing parabolic reflectors for antennas and solar concentrators, where the vertex serves as the focal point.
  • Economics: Optimizing production and pricing strategies, where the vertex indicates the point of maximum profit or minimum cost.
  • Computer Graphics: Creating smooth curves and shapes, where quadratic functions are used to define Bezier curves.

Conclusion

Quadratic functions, with their parabolic graphs and distinctive properties, stand as fundamental mathematical tools with far-reaching applications. The vertex, as the parabola's extreme point, plays a pivotal role in understanding and interpreting quadratic functions. By mastering the concepts and techniques discussed in this article, one can gain a deeper appreciation for the power and versatility of quadratic functions in modeling and solving real-world problems.

The vertex of a quadratic function is a critical point that reveals key information about the parabola's behavior, including its maximum or minimum value and axis of symmetry. Whether you're a student grappling with quadratic equations or a professional seeking to model real-world phenomena, mastering the process of finding the vertex is essential. This guide provides a comprehensive, step-by-step approach to finding the vertex of a quadratic function, equipping you with the knowledge and skills to tackle any quadratic equation.

Understanding the General and Vertex Forms

Before diving into the steps, let's recap the two primary forms of a quadratic function:

  • General Form: f(x) = ax^2 + bx + c
  • Vertex Form: f(x) = a(x - h)^2 + k

The general form is the standard representation, while the vertex form explicitly reveals the vertex coordinates (h, k). Our goal is often to transform the general form into vertex form to readily identify the vertex.

Step-by-Step Process to Find the Vertex

Here's a detailed guide to finding the vertex of a quadratic function:

Step 1: Identify the Coefficients

Start by identifying the coefficients 'a', 'b', and 'c' from the general form of the quadratic function:

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

These coefficients are the numerical values that multiply the respective terms. For instance, in the quadratic function f(x) = 2x^2 - 5x + 3, the coefficients are a = 2, b = -5, and c = 3.

Step 2: Calculate the x-coordinate (h) of the Vertex

The x-coordinate (h) of the vertex is determined using the vertex formula:

h = -b / 2a

Substitute the values of 'a' and 'b' that you identified in Step 1 into this formula and perform the calculation. This will give you the x-coordinate of the vertex.

For example, using the same quadratic function f(x) = 2x^2 - 5x + 3, we have:

h = -(-5) / (2 * 2) = 5 / 4

Step 3: Calculate the y-coordinate (k) of the Vertex

To find the y-coordinate (k) of the vertex, substitute the calculated value of 'h' back into the original quadratic function:

k = f(h) = a(h)^2 + b(h) + c

Evaluate the expression to obtain the y-coordinate of the vertex.

Continuing with our example, we substitute h = 5/4 into f(x) = 2x^2 - 5x + 3:

k = f(5/4) = 2(5/4)^2 - 5(5/4) + 3 = -1/8

Step 4: Express the Vertex Coordinates

The vertex coordinates are represented as (h, k). You have now calculated both 'h' and 'k', so you can express the vertex coordinates as an ordered pair.

In our example, the vertex coordinates are (5/4, -1/8).

Alternative Method: Completing the Square

An alternative method to find the vertex involves completing the square. This technique transforms the general form into vertex form, directly revealing the vertex coordinates.

Step 1: Group the x terms

Group the terms containing 'x' together:

f(x) = a(x^2 + (b/a)x) + c

Step 2: Complete the Square

Take half of the coefficient of the 'x' term (b/a), square it, and add and subtract it inside the parenthesis:

f(x) = a(x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2) + c

Step 3: Rewrite as a Squared Term

Rewrite the expression inside the parenthesis as a squared term:

f(x) = a(x + b/2a)^2 - a(b/2a)^2 + c

Step 4: Simplify and Express in Vertex Form

Simplify the expression and express the quadratic function in vertex form:

f(x) = a(x - h)^2 + k

where h = -b/2a and k = c - a(b/2a)^2.

Conclusion

Finding the vertex of a quadratic function is a fundamental skill in mathematics and has wide-ranging applications. Whether you choose to use the vertex formula or complete the square, this step-by-step guide provides you with the tools to confidently determine the vertex coordinates and gain valuable insights into the behavior of quadratic functions. By mastering this process, you'll be well-equipped to tackle quadratic equations and apply them effectively in various contexts.