How To Find The Axis Of Symmetry Of F(x)=-2(x+3)^2-5
Finding the axis of symmetry is a fundamental concept in understanding quadratic functions. In this article, we will delve into determining the axis of symmetry for the quadratic function f(x) = -2(x + 3)² - 5. We will explore the standard form of a quadratic equation, identify key parameters, and then apply these insights to pinpoint the axis of symmetry. This process is crucial not only for solving mathematical problems but also for grasping the behavior and graphical representation of quadratic functions.
Understanding Quadratic Functions and Their Forms
Quadratic functions are polynomial functions of the second degree, generally 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. The vertex of the parabola, which is either the minimum or maximum point of the function, plays a crucial role in determining its axis of symmetry.
The standard form or vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex of the parabola. This form is particularly useful because it directly reveals the vertex, which is essential for identifying the axis of symmetry. The parameter a in this form determines the direction and the “width” of the parabola. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The magnitude of a affects the steepness of the curve; a larger magnitude results in a steeper parabola, while a smaller magnitude leads to a flatter parabola.
In the given function, f(x) = -2(x + 3)² - 5, we can easily identify the parameters in the standard form. Here, a = -2, h = -3, and k = -5. This means the parabola opens downwards (since a is negative) and the vertex of the parabola is at the point (-3, -5). Understanding these parameters is the first step in finding the axis of symmetry, which will be a vertical line passing through the vertex.
Identifying the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. For a quadratic function in the standard form f(x) = a(x - h)² + k, the axis of symmetry is given by the equation x = h, where h is the x-coordinate of the vertex. This vertical line represents the mirror image of the parabola, meaning that if you were to fold the parabola along this line, the two halves would perfectly overlap. The axis of symmetry is a critical feature of a parabola because it provides a clear visual reference for the symmetry of the function.
In our function, f(x) = -2(x + 3)² - 5, we have already identified the vertex as (-3, -5). The h value, which is the x-coordinate of the vertex, is -3. Therefore, the equation of the axis of symmetry is x = -3. This means that the vertical line passing through x = -3 is the line of symmetry for the parabola represented by the given quadratic function. All points on the parabola have a corresponding point on the opposite side of this line, maintaining the symmetry of the curve.
To further illustrate this, consider a point on the parabola. If we move a certain distance horizontally from the axis of symmetry to this point, there will be another point on the parabola at the same vertical level but on the opposite side of the axis of symmetry, and the same horizontal distance away. This property of symmetry simplifies the process of graphing the parabola and understanding its behavior. Knowing the axis of symmetry allows us to predict how the function will behave on either side of the vertex, making it a valuable tool in analyzing quadratic functions.
Step-by-Step Determination of the Axis of Symmetry
To determine the axis of symmetry for the function f(x) = -2(x + 3)² - 5, we follow a straightforward, step-by-step process. This methodical approach ensures accuracy and clarity in the solution. The process involves recognizing the standard form of the quadratic equation, identifying the vertex, and then stating the equation of the axis of symmetry.
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Recognize the Standard Form: The given function, f(x) = -2(x + 3)² - 5, is already presented in the standard form f(x) = a(x - h)² + k. This form is particularly advantageous because it immediately reveals the vertex of the parabola. Recognizing this form is the crucial first step in determining the axis of symmetry.
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Identify the Vertex: In the standard form, the vertex of the parabola is given by the coordinates (h, k). By comparing the given function with the standard form, we can identify h and k. In our case, we have a = -2, and the term inside the parenthesis is (x + 3), which can be rewritten as (x - (-3)). Thus, h = -3. The constant term outside the parenthesis is k = -5. Therefore, the vertex of the parabola is at the point (-3, -5).
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State the Equation of the Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. Its equation is given by x = h, where h is the x-coordinate of the vertex. Since we have identified h as -3, the equation of the axis of symmetry for the given function is x = -3. This vertical line divides the parabola into two symmetrical halves, providing a clear visual representation of the function’s symmetry.
By following these steps, we have accurately determined that the axis of symmetry for the function f(x) = -2(x + 3)² - 5 is x = -3. This methodical approach can be applied to any quadratic function presented in standard form, making it a valuable tool in understanding and analyzing these functions.
Analyzing the Given Options
Having determined that the axis of symmetry for the function f(x) = -2(x + 3)² - 5 is x = -3, we can now analyze the provided options to select the correct one. This involves a direct comparison between our calculated result and the given choices.
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Option A: x = -5
This option is incorrect. The value -5 corresponds to the y-coordinate (k) of the vertex, not the x-coordinate (h), which determines the axis of symmetry. Choosing this option would indicate a misunderstanding of the roles of h and k in the standard form of a quadratic equation.
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Option B: x = 5
This option is also incorrect. The value 5 is the additive inverse of a component within the function but does not accurately represent the axis of symmetry. This choice might arise from overlooking the negative sign within the standard form equation or misinterpreting the transformation applied to the parabola.
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Option C: x = -3
This option is the correct answer. As we determined earlier, the axis of symmetry is given by x = h, and in this case, h = -3. Thus, the correct axis of symmetry is indeed the vertical line x = -3. This choice accurately reflects the symmetry of the parabola around the vertex.
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Option D: x = 3
This option is incorrect. While the value 3 is related to the horizontal shift of the parabola, it has the wrong sign. This mistake could occur from neglecting the negative sign within the standard form of the quadratic equation, leading to an incorrect determination of the axis of symmetry.
By carefully comparing our calculated axis of symmetry with the given options, we can confidently select Option C: x = -3 as the correct answer. This analysis not only confirms our solution but also reinforces the importance of accurately identifying the parameters in the standard form of a quadratic function.
Conclusion
In conclusion, to determine the axis of symmetry for the function f(x) = -2(x + 3)² - 5, we identified the function as being in standard form, recognized the vertex coordinates as (-3, -5), and applied the principle that the axis of symmetry is given by the equation x = h. Therefore, the axis of symmetry for the given function is x = -3. This understanding is crucial for effectively analyzing and graphing quadratic functions. The process involves recognizing the standard form, correctly identifying the vertex, and then applying the formula for the axis of symmetry. This ensures an accurate determination and a deeper understanding of the function’s behavior and graphical representation.
