Identifying Quadratic Functions With Axis Of Symmetry X=-2

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Determining the axis of symmetry is a fundamental concept in understanding quadratic functions. The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical halves. For a quadratic function in the standard form of f(x) = ax² + bx + c, the axis of symmetry can be found using the formula x = -b / 2a. This article will explore how to identify quadratic functions that have an axis of symmetry at x = -2 by applying this formula and analyzing the given functions.

Understanding the Axis of Symmetry

Before diving into the specific functions, it's essential to grasp the concept of the axis of symmetry. The axis of symmetry is a line that cuts a parabola into two mirror-image halves. This line always passes through the vertex, which is the highest or lowest point on the parabola. The x-coordinate of the vertex is precisely where the axis of symmetry lies. In mathematical terms, for a quadratic equation f(x) = ax² + bx + c, the axis of symmetry is a vertical line defined by the equation x = -b / 2a. This formula is derived from completing the square or using calculus to find the minimum or maximum point of the quadratic function. Understanding this formula allows us to quickly determine the axis of symmetry for any quadratic function given in standard form. The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0), while b and c affect the position of the vertex and, consequently, the axis of symmetry. The c value represents the y-intercept of the parabola, but it does not directly influence the axis of symmetry. Thus, to find functions with a specific axis of symmetry, we primarily focus on the relationship between the coefficients a and b.

Analyzing the Given Functions

We are tasked with identifying which of the following quadratic functions have an axis of symmetry at x = -2. To do this, we will use the formula x = -b / 2a for each function and check if it equals -2.

1. f(x) = x² + 4x + 3

Let's start with the first function, f(x) = x² + 4x + 3. In this case, a = 1 and b = 4. Plugging these values into the axis of symmetry formula, we get:

x = -b / 2a = -4 / (2 * 1) = -4 / 2 = -2

Since the axis of symmetry is x = -2, this function meets our criteria. The graph of this function is a parabola opening upwards, and its vertex lies on the line x = -2. This means that the function is symmetric around this vertical line. The c value, which is 3, indicates the y-intercept of the parabola, but it does not affect the axis of symmetry. Therefore, the axis of symmetry is solely determined by the relationship between the coefficients a and b. In this particular case, the negative of b divided by twice a gives us -2, confirming that this function has the desired symmetry.

2. f(x) = x² - 4x - 5

Next, we consider the function f(x) = x² - 4x - 5. Here, a = 1 and b = -4. Applying the formula:

x = -b / 2a = -(-4) / (2 * 1) = 4 / 2 = 2

The axis of symmetry for this function is x = 2, which does not match our target of x = -2. This parabola also opens upwards, but its vertex is located at x = 2, indicating a symmetry around this different vertical line. The negative sign in front of the 4 in the original equation results in a positive value when calculating the axis of symmetry, leading to the axis being at x = 2 instead of x = -2. This illustrates how the sign of the b coefficient significantly impacts the location of the axis of symmetry. Consequently, this function does not meet the required condition.

3. f(x) = x² + 6x + 2

Now, let’s examine f(x) = x² + 6x + 2. In this function, a = 1 and b = 6. Using the formula:

x = -b / 2a = -6 / (2 * 1) = -6 / 2 = -3

The axis of symmetry for this function is x = -3, which is also not x = -2. This parabola opens upwards, and its vertex is situated on the line x = -3. This deviation from the target axis of symmetry is due to the larger value of b in relation to a. The axis of symmetry shifts further to the left as b increases when a remains constant. Therefore, this function does not satisfy the condition of having an axis of symmetry at x = -2.

4. f(x) = -2x² - 8x + 1

For the function f(x) = -2x² - 8x + 1, we have a = -2 and b = -8. Plugging these values into the formula:

x = -b / 2a = -(-8) / (2 * -2) = 8 / -4 = -2

This function has an axis of symmetry at x = -2, thus fulfilling our requirement. The graph of this function is a parabola opening downwards since a is negative, and its vertex lies on the line x = -2. The symmetry around this line confirms that this function fits the criteria. The coefficients a and b interplay in a way that the negative of b divided by twice a results in -2, which aligns with the target axis of symmetry. This demonstrates that even with a negative leading coefficient, the formula x = -b / 2a accurately predicts the axis of symmetry.

5. f(x) = -2x² + 8x - 2

Lastly, we analyze f(x) = -2x² + 8x - 2. Here, a = -2 and b = 8. Applying the formula:

x = -b / 2a = -8 / (2 * -2) = -8 / -4 = 2

The axis of symmetry for this function is x = 2, which is not x = -2. This parabola opens downwards, but its vertex is located at x = 2, indicating symmetry around this vertical line instead. The positive b coefficient, combined with the negative a coefficient, leads to a positive axis of symmetry. Therefore, this function does not have the required axis of symmetry.

Conclusion

In summary, by applying the formula x = -b / 2a to each function, we determined that the functions f(x) = x² + 4x + 3 and f(x) = -2x² - 8x + 1 have an axis of symmetry at x = -2. Understanding how to calculate the axis of symmetry is crucial for analyzing quadratic functions and their graphs. This skill helps in identifying key features of parabolas, such as the vertex and the direction of opening, which are fundamental concepts in algebra and calculus.

By systematically analyzing each function, we've highlighted the importance of the relationship between the coefficients a and b in determining the axis of symmetry. Functions with the same axis of symmetry share a specific ratio between these coefficients, allowing for a deeper understanding of quadratic function behavior. This exercise reinforces the application of algebraic formulas and their significance in graphical representations of functions.