Vertex And Minimum Value Of Quadratic Function F(x)=2(x+2)^2-9

In this article, we will delve into the process of identifying the vertex of a quadratic function and determining whether it represents a minimum or maximum value. Specifically, we will analyze the function $f(x) = 2(x + 2)^2 - 9$ to find its vertex and ascertain its minimum or maximum value. Understanding these concepts is crucial in various mathematical and real-world applications, from optimizing quantities to modeling parabolic trajectories.

Understanding Quadratic Functions and Vertex Form

Quadratic functions are polynomial functions of degree two, generally expressed in the standard form as $f(x) = ax^2 + bx + c$, 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 upwards if $a > 0$ and downwards if $a < 0$. The vertex of the parabola is the point where the parabola changes direction, representing either the minimum or maximum value of the function.

A particularly useful form for analyzing quadratic functions is the vertex form, given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. The coefficient $a$ determines the direction and width of the parabola. If $a > 0$, the parabola opens upwards, and the vertex represents the minimum value of the function. Conversely, if $a < 0$, the parabola opens downwards, and the vertex represents the maximum value of the function. The absolute value of $a$ dictates the parabola's width; a larger $|a|$ results in a narrower parabola, while a smaller $|a|$ leads to a wider parabola.

The vertex form provides a direct way to identify the vertex and determine the minimum or maximum value of the quadratic function. By rewriting a quadratic function in vertex form, we can easily read off the coordinates of the vertex and the direction in which the parabola opens. This form is especially valuable for graphing quadratic functions and solving optimization problems.

Analyzing the Given Function: $f(x) = 2(x + 2)^2 - 9$

Now, let's focus on the given function: $f(x) = 2(x + 2)^2 - 9$. This function is already presented in vertex form, which makes our task of identifying the vertex and minimum/maximum value significantly simpler. By comparing this function with the general vertex form $f(x) = a(x - h)^2 + k$, we can directly extract the values of $a$, $h$, and $k$.

In our case, we have $a = 2$, $h = -2$, and $k = -9$. The vertex of the parabola is therefore given by the coordinates $(h, k) = (-2, -9)$. Since $a = 2$ is positive, the parabola opens upwards, indicating that the vertex represents the minimum value of the function. The minimum value of the function is the y-coordinate of the vertex, which is $k = -9$.

To further illustrate this, consider the term $(x + 2)^2$. Since any squared term is non-negative, $(x + 2)^2$ is always greater than or equal to zero. When $x = -2$, this term becomes zero, and the function attains its minimum value. For any other value of $x$, $(x + 2)^2$ will be positive, causing the function value to be greater than -9. This confirms that the vertex $(-2, -9)$ indeed corresponds to the minimum point on the graph of the function.

Determining the Vertex and Minimum Value

Based on our analysis, the vertex of the function $f(x) = 2(x + 2)^2 - 9$ is (-2, -9). Since the coefficient $a = 2$ is positive, the parabola opens upwards, and the vertex represents the minimum value of the function. The minimum value of the function is the y-coordinate of the vertex, which is -9.

Therefore, the correct answer is: Vertex: (-2, -9), Minimum: -9. This means that the function reaches its lowest point at the coordinates (-2, -9) on the graph, and the minimum value of the function is -9. Understanding how to identify the vertex and minimum or maximum value of a quadratic function is a fundamental skill in algebra and calculus, with applications in optimization problems and curve sketching.

Graphing the Function and Visualizing the Vertex

To further solidify our understanding, let's visualize the graph of the function $f(x) = 2(x + 2)^2 - 9$. We know that the parabola opens upwards and has its vertex at (-2, -9). To sketch the graph, we can find a few additional points. For example, let's find the y-intercept by setting $x = 0$:

$f(0) = 2(0 + 2)^2 - 9 = 2(4) - 9 = 8 - 9 = -1$

So, the y-intercept is (0, -1). Due to the symmetry of the parabola, there will be another point at $x = -4$ with the same y-coordinate. Let's verify:

$f(-4) = 2(-4 + 2)^2 - 9 = 2(-2)^2 - 9 = 2(4) - 9 = 8 - 9 = -1$

Thus, the point (-4, -1) is also on the graph. With the vertex (-2, -9) and these two additional points (0, -1) and (-4, -1), we can sketch the parabola. The graph clearly shows that the vertex is the lowest point on the curve, confirming that it represents the minimum value of the function.

Graphing the function provides a visual representation of the algebraic analysis we performed earlier. It allows us to see how the vertex and the direction of the parabola are related to the coefficients in the quadratic function. This visual understanding can be invaluable when solving problems involving quadratic functions and their applications.

Applications of Vertex and Minimum/Maximum Values

The concept of the vertex and minimum/maximum values of quadratic functions has numerous applications in various fields. Here are a few examples:

1. Optimization Problems: Quadratic functions are frequently used to model optimization problems, where we aim to find the maximum or minimum value of a certain quantity. For example, we might want to maximize the area of a rectangular garden given a fixed amount of fencing. The area can often be expressed as a quadratic function of the dimensions of the garden, and finding the vertex allows us to determine the dimensions that yield the maximum area.

2. Projectile Motion: The trajectory of a projectile, such as a ball thrown in the air, can be modeled using a quadratic function. The vertex of the parabola represents the highest point reached by the projectile, and the minimum value (if the parabola opens downwards) can represent the point where the projectile hits the ground. Understanding the vertex allows us to determine the maximum height and range of the projectile.

3. Engineering Design: In engineering, quadratic functions are used to model various phenomena, such as the shape of suspension cables on bridges or the deflection of beams under load. Determining the vertex and minimum/maximum values helps engineers design structures that can withstand specific forces and loads.

4. Economics and Business: Quadratic functions can be used to model cost, revenue, and profit functions in business. Finding the vertex allows businesses to determine the optimal production level or price that maximizes profit or minimizes cost.

The ability to identify the vertex and minimum/maximum values of quadratic functions is a powerful tool with wide-ranging applications in various disciplines. By mastering this concept, you can solve a variety of real-world problems and gain a deeper understanding of the mathematical principles that govern them.

Conclusion

In summary, we have successfully identified the vertex and determined the minimum value of the function $f(x) = 2(x + 2)^2 - 9$. By recognizing the vertex form of the quadratic function, we directly extracted the vertex coordinates as (-2, -9). The positive coefficient of the squared term indicated that the parabola opens upwards, and the vertex represents the minimum value of the function, which is -9. We further reinforced our understanding by visualizing the graph of the function and discussing various applications of the vertex and minimum/maximum values in real-world scenarios.

Understanding how to analyze quadratic functions and identify their key features, such as the vertex and minimum/maximum values, is essential for success in mathematics and its applications. By practicing these skills and applying them to various problems, you can develop a strong foundation in quadratic functions and their role in modeling and solving real-world challenges.