Finding The Range Of F(x) = 3x² + 6x - 8 A Step-by-Step Guide

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Determining the range of a function is a fundamental concept in mathematics, particularly when dealing with quadratic functions. The range represents the set of all possible output values (y-values) that the function can produce. In this comprehensive guide, we will delve deep into the process of finding the range of the quadratic function f(x) = 3x² + 6x - 8. We will explore various methods, including completing the square and using the vertex formula, to arrive at the correct answer. This article aims to provide a clear and thorough understanding, ensuring you grasp the underlying principles and can confidently tackle similar problems.

Defining the Range of a Function

Before we dive into the specifics of the given function, let's first establish a clear understanding of what the range of a function means. In simple terms, the range is the set of all possible y-values that the function can output when you plug in different x-values. For a quadratic function, which graphs as a parabola, the range is closely related to the vertex of the parabola. The vertex represents either the minimum or maximum point of the parabola, and this point plays a crucial role in determining the range. Understanding the range helps us define the boundaries of the function's output and provides valuable insights into its behavior.

Exploring Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards depending on the sign of the coefficient a. If a is positive, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if a is negative, the parabola opens downwards, and the vertex represents the maximum point. Recognizing this fundamental property is essential for determining the range of any quadratic function.

The vertex of a parabola is the point where the function reaches its minimum or maximum value. The x-coordinate of the vertex can be found using the formula x = -b / 2a. Once you have the x-coordinate, you can substitute it back into the function to find the y-coordinate of the vertex, which represents the minimum or maximum value of the function. This value is the key to determining the range. The parabola's symmetry about the vertical line passing through the vertex ensures that all other y-values will be greater than or equal to (if the parabola opens upwards) or less than or equal to (if the parabola opens downwards) the y-value of the vertex.

Analyzing the Function f(x) = 3x² + 6x - 8

Now, let's focus on the specific quadratic function given: f(x) = 3x² + 6x - 8. Our goal is to find the range of this function, which means we need to determine all the possible y-values it can produce. To do this, we'll use the techniques discussed earlier, focusing on finding the vertex of the parabola. The coefficients of this quadratic function are a = 3, b = 6, and c = -8. Since a is positive (3 > 0), we know that the parabola opens upwards, meaning the vertex represents the minimum point of the function. This tells us that the range will be all y-values greater than or equal to the y-coordinate of the vertex.

Method 1: Completing the Square

One effective method to find the vertex is by completing the square. This technique involves rewriting the quadratic expression in vertex form, which is f(x) = a(x - h)² + k, where (h, k) are the coordinates of the vertex. This form directly reveals the vertex of the parabola, making it easy to determine the range.

Let's apply completing the square to our function f(x) = 3x² + 6x - 8:

  1. Factor out the coefficient of the x² term (which is 3) from the first two terms: f(x) = 3(x² + 2x) - 8
  2. Complete the square inside the parentheses. To do this, take half of the coefficient of the x term (which is 2), square it (1² = 1), and add it inside the parentheses. To maintain the equality, we must also subtract 3 times this value outside the parentheses (since we're multiplying the parentheses by 3): f(x) = 3(x² + 2x + 1) - 8 - 3(1)
  3. Rewrite the expression inside the parentheses as a perfect square: f(x) = 3(x + 1)² - 8 - 3
  4. Simplify the expression: f(x) = 3(x + 1)² - 11

Now the function is in vertex form, f(x) = 3(x + 1)² - 11. We can see that the vertex of the parabola is at the point (-1, -11). Since the parabola opens upwards, the minimum value of the function is -11. Therefore, the range of the function is all y-values greater than or equal to -11.

Method 2: Using the Vertex Formula

Another way to find the vertex is by using the vertex formula. As mentioned earlier, the x-coordinate of the vertex is given by x = -b / 2a. Once we have the x-coordinate, we can substitute it back into the function to find the y-coordinate.

For our function f(x) = 3x² + 6x - 8, we have a = 3 and b = 6. Let's apply the vertex formula:

  1. Find the x-coordinate of the vertex: x = -b / 2a = -6 / (2 * 3) = -6 / 6 = -1
  2. Substitute x = -1 back into the function to find the y-coordinate: f(-1) = 3(-1)² + 6(-1) - 8 = 3 - 6 - 8 = -11

Thus, the vertex of the parabola is (-1, -11), which confirms the result we obtained by completing the square. Again, since the parabola opens upwards, the minimum value of the function is -11, and the range is all y-values greater than or equal to -11.

Determining the Range and Selecting the Correct Option

Both methods, completing the square and using the vertex formula, have led us to the same conclusion: the vertex of the parabola is (-1, -11), and the parabola opens upwards. This means the minimum value of the function is -11. Therefore, the range of the function f(x) = 3x² + 6x - 8 is all y-values greater than or equal to -11.

Looking at the options provided:

A. {y | y ≥ -1} B. {y | y ≤ -1} C. {y | y ≥ -11} D. {y | y ≤ -11}

The correct option is C. {y | y ≥ -11}, as it accurately represents the range of the function.

Conclusion

In this comprehensive guide, we've thoroughly explored the process of finding the range of the quadratic function f(x) = 3x² + 6x - 8. We've discussed the fundamental concept of the range of a function, examined the properties of quadratic functions and parabolas, and applied two different methods – completing the square and using the vertex formula – to determine the vertex of the parabola. Both methods confirmed that the vertex is (-1, -11), and since the parabola opens upwards, the range of the function is y ≥ -11. This detailed explanation, combined with the step-by-step solutions, equips you with a strong understanding of how to find the range of quadratic functions. Remember to always consider the sign of the coefficient a and the location of the vertex, as these are the key elements in determining the range.