Finding The Axis Of Symmetry For Quadratic Function F(x) = -x^2 + X + 6

In mathematics, especially within algebra, quadratic functions play a pivotal role. These functions, characterized by their parabolic curves, are not only visually intriguing but also deeply practical, finding applications in physics, engineering, economics, and computer graphics. This article delves into a specific quadratic function, $f(x) = -x^2 + x + 6$, to explore its properties, particularly focusing on how to determine its axis of symmetry.

The axis of symmetry is a crucial characteristic of a parabola. It's a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Understanding the axis of symmetry helps in graphing the function, identifying the maximum or minimum value, and solving related problems. In this comprehensive exploration, we will analyze the given table of values, calculate the missing values $a$, $b$, and $c$, and then determine the axis of symmetry for the quadratic function $f(x) = -x^2 + x + 6$.

Analyzing the Quadratic Function $f(x) = -x^2 + x + 6$

The quadratic function $f(x) = -x^2 + x + 6$ is a polynomial of degree two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a e 0$. In our case, $a = -1$, $b = 1$, and $c = 6$. The negative coefficient of the $x^2$ term indicates that the parabola opens downward, meaning it has a maximum value.

Determining the Missing Values: $a$, $b$, and $c$

The provided table gives us several $x$-values and their corresponding function values $f(x)$. Let's use this information to compute the missing values $a$, $b$, and $c$. These values are crucial for understanding the behavior and graph of the quadratic function.

1. Calculating a: When $x = -2$, $f(x) = a$. Substituting $x = -2$ into the function:

   $f(-2) = -(-2)^2 + (-2) + 6$

   $f(-2) = -(4) - 2 + 6$

   $f(-2) = -4 - 2 + 6$

   $f(-2) = 0$

   Therefore, $a = 0$.

2. Calculating b: When $x = 0$, $f(x) = b$. Substituting $x = 0$ into the function:

   $f(0) = -(0)^2 + (0) + 6$

   $f(0) = 0 + 0 + 6$

   $f(0) = 6$

   Therefore, $b = 6$.

3. Calculating c: When $x = 2$, $f(x) = c$. Substituting $x = 2$ into the function:

   $f(2) = -(2)^2 + (2) + 6$

   $f(2) = -4 + 2 + 6$

   $f(2) = 4$

   Therefore, $c = 4$.

Now that we have calculated the missing values, we can update our table:

Understanding the Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the vertex of the parabola, which is the point where the function reaches its maximum (or minimum) value. For a quadratic function in the form $f(x) = ax^2 + bx + c$, the axis of symmetry can be found using the formula:

x = -rac{b}{2a}

This formula is derived from completing the square or using calculus to find the critical point of the quadratic function. It provides a straightforward method to determine the axis of symmetry without graphing the function.

Calculating the Axis of Symmetry for $f(x) = -x^2 + x + 6$

Using the formula for the axis of symmetry, we can plug in the values of $a$ and $b$ from our quadratic function $f(x) = -x^2 + x + 6$. Here, $a = -1$ and $b = 1$.

x = -rac{b}{2a} = -rac{1}{2(-1)} = -rac{1}{-2} = rac{1}{2}

Therefore, the axis of symmetry for the function $f(x) = -x^2 + x + 6$ is x = rac{1}{2}.

This means that the vertical line x = rac{1}{2} divides the parabola into two symmetrical halves. The vertex of the parabola lies on this line, and the function values are mirrored across this line.

Verifying the Axis of Symmetry

To verify our calculation, we can look at the table of values and observe the symmetry around x = rac{1}{2}. The table shows the following points:

• $(-2, 0)$
• $(-1, 4)$
• $(0, 6)$
• $(1, 6)$
• $(2, 4)$

Notice how the function values are symmetrical around the vertex. The points $(0, 6)$ and $(1, 6)$ have the same y-value, and the points $(-1, 4)$ and $(2, 4)$ also share the same y-value. This symmetry confirms that the axis of symmetry lies between these points.

The vertex of the parabola occurs at the point where x = rac{1}{2}. To find the y-coordinate of the vertex, we substitute x = rac{1}{2} into the function:

f(rac{1}{2}) = -(rac{1}{2})^2 + rac{1}{2} + 6

f(rac{1}{2}) = -rac{1}{4} + rac{1}{2} + 6

f(rac{1}{2}) = -rac{1}{4} + rac{2}{4} + rac{24}{4}

f(rac{1}{2}) = rac{25}{4}

So, the vertex of the parabola is at the point (rac{1}{2}, rac{25}{4}). This point lies on the axis of symmetry, further confirming our result.

Graphing the Quadratic Function

Graphing the quadratic function provides a visual representation of the axis of symmetry. The parabola opens downwards since the coefficient of $x^2$ is negative. The vertex is the highest point on the graph, and the axis of symmetry is the vertical line that passes through this vertex.

To graph the function, we can plot the points from the table and the vertex (rac{1}{2}, rac{25}{4}). The symmetry around the axis x = rac{1}{2} will be evident in the graph.

Key Features of the Graph

• Vertex: The vertex is the point (rac{1}{2}, rac{25}{4}), which is the maximum point of the parabola.
• Axis of Symmetry: The vertical line x = rac{1}{2} divides the parabola into two symmetrical halves.
• Y-intercept: The y-intercept is the point where the graph intersects the y-axis. This occurs when $x = 0$, so the y-intercept is $(0, 6)$.
• X-intercepts: The x-intercepts are the points where the graph intersects the x-axis. These are the solutions to the equation $f(x) = 0$. In this case, the x-intercepts can be found by solving $-x^2 + x + 6 = 0$. Factoring the quadratic equation, we get $-(x - 3)(x + 2) = 0$, so the x-intercepts are $x = 3$ and $x = -2$. The points are $(3, 0)$ and $(-2, 0)$.

Importance of the Axis of Symmetry

The axis of symmetry is not just a visual characteristic; it has practical implications in various fields. Understanding the axis of symmetry helps in solving real-world problems involving quadratic functions.

Applications in Physics

In physics, projectile motion can be modeled using quadratic functions. The axis of symmetry represents the time at which the projectile reaches its maximum height. By finding the axis of symmetry, we can determine when the projectile is at its highest point and how far it travels horizontally.

Applications in Engineering

In engineering, quadratic functions are used in the design of parabolic reflectors, such as satellite dishes and car headlights. The axis of symmetry is crucial in aligning the focus of the reflector, ensuring optimal performance.

Applications in Economics

In economics, quadratic functions can model cost and revenue curves. The axis of symmetry can help determine the production level that maximizes profit or minimizes cost.

Applications in Computer Graphics

In computer graphics, parabolas are used to create smooth curves and surfaces. Understanding the axis of symmetry is essential for manipulating and rendering these curves efficiently.

Conclusion

In summary, the axis of symmetry for the quadratic function $f(x) = -x^2 + x + 6$ is x = rac{1}{2}. We determined this by using the formula x = -rac{b}{2a}, where $a = -1$ and $b = 1$. We also verified our result by analyzing the table of values and observing the symmetry around x = rac{1}{2}.

The axis of symmetry is a fundamental property of parabolas, providing valuable information about the function's behavior and graph. It helps in identifying the vertex, understanding the symmetry, and solving practical problems in various fields. By mastering the concept of the axis of symmetry, we can gain a deeper understanding of quadratic functions and their applications.

Understanding quadratic functions and their properties, such as the axis of symmetry, is essential in mathematics and various real-world applications. By using the formula and verifying the results, we can confidently determine the axis of symmetry and apply this knowledge to solve practical problems.