Find The Range Of The Quadratic Function

Let's delve into the heart of quadratic functions and unravel the mystery of how to determine their range. Understanding the range of a quadratic function is essential for grasping its behavior and properties. This comprehensive guide will equip you with the knowledge and techniques to confidently find the range of any quadratic function you encounter.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two. It can be expressed in the general 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. The parabola opens upwards if a > 0 and downwards if a < 0.

Key Concepts for Finding the Range

Before diving into the methods for finding the range, it's crucial to understand these fundamental concepts:

• Vertex: The vertex is the point where the parabola changes direction. It's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). The x-coordinate of the vertex is given by -b/2a. The y-coordinate of the vertex is found by substituting the x-coordinate back into the original quadratic function.

• Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b/2a.

• Leading Coefficient (a): The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). It also influences the "width" of the parabola. A larger absolute value of a results in a narrower parabola.

• Minimum/Maximum Value: The y-coordinate of the vertex represents the minimum value of the function if the parabola opens upwards (a > 0) and the maximum value if the parabola opens downwards (a < 0).

Methods to Determine the Range of a Quadratic Function

There are several ways to find the range of a quadratic function. We'll explore the most common and effective methods:

1. Using the Vertex Form

The vertex form of a quadratic function is given by:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola. This form directly reveals the vertex and the direction of the parabola, making it easy to determine the range.

• If a > 0 (parabola opens upwards): The range is [k, ∞). This means the minimum value of the function is k, and the function extends to positive infinity.
• If a < 0 (parabola opens downwards): The range is (-∞, k]. This means the maximum value of the function is k, and the function extends to negative infinity.

Example:

Consider the quadratic function f(x) = 2(x - 1)² + 3.

Here, a = 2, h = 1, and k = 3. Since a > 0, the parabola opens upwards. Therefore, the range is [3, ∞).

Converting from Standard Form to Vertex Form:

If the quadratic function is given in standard form (f(x) = ax² + bx + c), you need to convert it to vertex form. This can be done by completing the square:

1. Factor out a from the x² and x terms:

   f(x) = a(x² + (b/a)x) + c

2. Complete the square inside the parentheses:

   Take half of the coefficient of the x term (b/a), square it ((b/2a)²), and add and subtract it inside the parentheses.

   f(x) = a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c

3. Rewrite the expression inside the parentheses as a squared term:

   f(x) = a((x + b/2a)² - (b/2a)²) + c

4. Distribute a and simplify:

   f(x) = a(x + b/2a)² - a(b/2a)² + c f(x) = a(x + b/2a)² - b²/4a + c f(x) = a(x + b/2a)² + (4ac - b²)/4a

Now, the function is in vertex form, where h = -b/2a and k = (4ac - b²)/4a.

2. Using the Discriminant

The discriminant (Δ) of a quadratic equation ax² + bx + c = 0 is given by:

Δ = b² - 4ac

The discriminant provides information about the nature of the roots of the quadratic equation. However, it can also be used to find the range of the quadratic function f(x) = ax² + bx + c.

• Finding the Vertex's y-coordinate (k): The y-coordinate of the vertex, k, which is crucial for determining the range, can be found using the discriminant:

  k = -Δ / 4a = -(b² - 4ac) / 4a = (4ac - b²) / 4a

  This formula is the same as the k value we derived when completing the square.

• Determining the Range:

  • If a > 0: The range is [k, ∞) = [-(b² - 4ac) / 4a, ∞)
  • If a < 0: The range is (-∞, k] = (-∞, -(b² - 4ac) / 4a]

Example:

Consider the quadratic function f(x) = -x² + 4x - 1.

Here, a = -1, b = 4, and c = -1.

1. Calculate the discriminant:

   Δ = b² - 4ac = 4² - 4(-1)(-1) = 16 - 4 = 12

2. Find the y-coordinate of the vertex (k):

   k = -Δ / 4a = -12 / (4 * -1) = -12 / -4 = 3

3. Determine the range:

   Since a < 0, the range is (-∞, k] = (-∞, 3].

3. Finding the Vertex Directly

This method involves directly calculating the coordinates of the vertex (h, k) and then determining the range based on the direction of the parabola.

• Find the x-coordinate of the vertex (h):

  h = -b / 2a

• Find the y-coordinate of the vertex (k):

  Substitute the value of h back into the original quadratic function:

  k = f(h) = a(-b/2a)² + b(-b/2a) + c

• Determine the Range:

  • If a > 0: The range is [k, ∞)
  • If a < 0: The range is (-∞, k]

Example:

Consider the quadratic function f(x) = x² - 6x + 5.

Here, a = 1, b = -6, and c = 5.

1. Find the x-coordinate of the vertex (h):

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

2. Find the y-coordinate of the vertex (k):

   k = f(3) = (3)² - 6(3) + 5 = 9 - 18 + 5 = -4

3. Determine the range:

   Since a > 0, the range is [-4, ∞).

Examples and Applications

Let's solidify our understanding with more examples:

Example 1: f(x) = 3x² + 12x - 1

• a = 3, b = 12, c = -1
• h = -b/2a = -12 / (2*3) = -2
• k = f(-2) = 3(-2)² + 12(-2) - 1 = 12 - 24 - 1 = -13
• Since a > 0, the range is [-13, ∞).

Example 2: f(x) = -2x² + 8x - 5

• a = -2, b = 8, c = -5
• h = -b/2a = -8 / (2*-2) = 2
• k = f(2) = -2(2)² + 8(2) - 5 = -8 + 16 - 5 = 3
• Since a < 0, the range is (-∞, 3].

Example 3: f(x) = (x + 3)² - 4

• This is already in vertex form: f(x) = a(x - h)² + k, where a = 1, h = -3, and k = -4.
• Since a > 0, the range is [-4, ∞).

Applications:

Understanding the range of a quadratic function is crucial in various applications, including:

• Optimization Problems: Determining the maximum profit, minimum cost, or optimal height of a projectile.
• Physics: Modeling projectile motion and determining the maximum height reached.
• Engineering: Designing parabolic reflectors or antennas.
• Economics: Modeling cost and revenue functions.

Common Mistakes to Avoid

• Confusing Range with Domain: The domain of a quadratic function is always all real numbers (-∞, ∞). The range is the set of possible output values (y-values).
• Incorrectly Identifying the Vertex: Ensure you calculate the vertex coordinates accurately using the correct formulas.
• Forgetting to Consider the Sign of 'a': The sign of 'a' determines whether the parabola opens upwards or downwards, which directly affects the range.
• Assuming the Range is Always All Real Numbers: This is only true for linear functions (degree 1), not quadratic functions.

Advanced Considerations

• Restricted Domains: If the domain of the quadratic function is restricted (e.g., only considering x values between 0 and 5), the range will also be affected. You'll need to evaluate the function at the endpoints of the restricted domain and compare those values with the vertex's y-coordinate to determine the range.
• Transformations of Quadratic Functions: Understanding how transformations (translations, reflections, stretches, and compressions) affect the graph of a quadratic function can help you quickly determine the range.

Practice Problems

To solidify your understanding, try these practice problems:

1. f(x) = x² + 2x + 3
2. f(x) = -x² + 6x - 8
3. f(x) = 2x² - 4x + 1
4. f(x) = -3(x - 1)² + 5
5. f(x) = (1/2)x² + x - 2

Answers:

1. [2, ∞)
2. (-∞, 1]
3. [-1, ∞)
4. (-∞, 5]
5. [-5/2, ∞)

Conclusion

Finding the range of a quadratic function is a fundamental skill in algebra and calculus. By understanding the key concepts, mastering the different methods (vertex form, discriminant, direct vertex calculation), and practicing with examples, you can confidently determine the range of any quadratic function you encounter. Remember to pay attention to the sign of the leading coefficient (a) and the vertex's y-coordinate, as these are crucial for determining the correct range. Keep practicing, and you'll become a pro at finding the range of quadratic functions!