How To Find The Range Of A Quadratic Function

Finding the range of a quadratic function is a fundamental skill in algebra. The range represents all possible output values (y-values) of the function. Understanding how to determine this range is crucial for analyzing and interpreting quadratic models in various real-world applications, from physics to economics. This article will guide you through the process, providing clear explanations, examples, and useful techniques to master this concept.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two. Its general form is:

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 opens either upwards or downwards, depending on the sign of a.

• If a > 0: The parabola opens upwards, and the vertex represents the minimum point of the function.
• If a < 0: The parabola opens downwards, and the vertex represents the maximum point of the function.

The vertex is a crucial point for determining the range of a quadratic function. Its coordinates are given by (h, k), where:

• h = -b / 2a
• k = f(h) (i.e., substitute h back into the original function)

The value of k directly relates to the minimum or maximum value of the function, which is essential for defining the range.

Steps to Find the Range of a Quadratic Function

Here’s a step-by-step process to find the range of any quadratic function:

1. Identify the coefficients a, b, and c.

2. Determine if the parabola opens upwards or downwards by looking at the sign of a.

3. Calculate the x-coordinate of the vertex (h) using the formula h = -b / 2a.

4. Calculate the y-coordinate of the vertex (k) by substituting h into the original function: k = f(h).

5. Write the range based on the direction of the parabola and the value of k.

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

Examples of Finding the Range

Let’s work through several examples to illustrate the process.

Example 1: f(x) = x² - 4x + 3

1. Identify coefficients: a = 1, b = -4, c = 3.
2. Direction of parabola: Since a = 1 > 0, the parabola opens upwards.
3. x-coordinate of vertex: h = -b / 2a = -(-4) / (2 * 1) = 4 / 2 = 2.
4. y-coordinate of vertex: k = f(2) = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1.
5. Range: Since the parabola opens upwards and the vertex is at (2, -1), the range is [-1, ∞).

Example 2: g(x) = -2x² + 8x - 5

1. Identify coefficients: a = -2, b = 8, c = -5.
2. Direction of parabola: Since a = -2 < 0, the parabola opens downwards.
3. x-coordinate of vertex: h = -b / 2a = -8 / (2 * -2) = -8 / -4 = 2.
4. y-coordinate of vertex: k = g(2) = -2(2)² + 8(2) - 5 = -8 + 16 - 5 = 3.
5. Range: Since the parabola opens downwards and the vertex is at (2, 3), the range is (-∞, 3].

Example 3: h(x) = 3x² + 6x + 1

1. Identify coefficients: a = 3, b = 6, c = 1.
2. Direction of parabola: Since a = 3 > 0, the parabola opens upwards.
3. x-coordinate of vertex: h = -b / 2a = -6 / (2 * 3) = -6 / 6 = -1.
4. y-coordinate of vertex: k = h(-1) = 3(-1)² + 6(-1) + 1 = 3 - 6 + 1 = -2.
5. Range: Since the parabola opens upwards and the vertex is at (-1, -2), the range is [-2, ∞).

Example 4: p(x) = -x² - 2x + 4

1. Identify coefficients: a = -1, b = -2, c = 4.
2. Direction of parabola: Since a = -1 < 0, the parabola opens downwards.
3. x-coordinate of vertex: h = -b / 2a = -(-2) / (2 * -1) = 2 / -2 = -1.
4. y-coordinate of vertex: k = p(-1) = -(-1)² - 2(-1) + 4 = -1 + 2 + 4 = 5.
5. Range: Since the parabola opens downwards and the vertex is at (-1, 5), the range is (-∞, 5].

Completing the Square Method

Another method to find the range of a quadratic function is by completing the square. This method rewrites the quadratic function in vertex form, which directly reveals the vertex coordinates and, consequently, the range.

The vertex form of a quadratic function is:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola.

Here’s how to complete 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 (which is 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 perfect square trinomial:

   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

5. Simplify to vertex form:

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

From this form, you can easily identify the vertex as (-b/2a, c - a(b/2a)²). Notice that h = -b/2a and k = c - a(b/2a)².

Example Using Completing the Square: f(x) = 2x² - 8x + 5

1. Factor out a (which is 2):

   f(x) = 2(x² - 4x) + 5

2. Complete the square: Half of -4 is -2, and (-2)² = 4. Add and subtract 4 inside the parentheses:

   f(x) = 2(x² - 4x + 4 - 4) + 5

3. Rewrite the perfect square trinomial:

   f(x) = 2((x - 2)² - 4) + 5

4. Distribute and simplify:

   f(x) = 2(x - 2)² - 8 + 5

5. Vertex form:

   f(x) = 2(x - 2)² - 3

The vertex is (2, -3). Since a = 2 > 0, the parabola opens upwards, and the range is [-3, ∞).

Domain Restrictions and Their Impact on the Range

In some cases, the domain of a quadratic function may be restricted. This means that the function is only defined for certain values of x. When the domain is restricted, the range can be affected, as the function may not reach its absolute minimum or maximum value.

To find the range with a restricted domain, follow these steps:

1. Find the vertex of the quadratic function as usual.
2. Evaluate the function at the endpoints of the domain.
3. Compare the y-values of the vertex and the endpoints.
4. Determine the highest and lowest y-values.
5. Write the range using these highest and lowest y-values.

Example with a Restricted Domain: f(x) = x² - 2x + 3, for 0 ≤ x ≤ 3

1. Find the vertex:
   • h = -b / 2a = -(-2) / (2 * 1) = 1
   • k = f(1) = (1)² - 2(1) + 3 = 1 - 2 + 3 = 2 The vertex is (1, 2).
2. Evaluate the function at the endpoints:
   • f(0) = (0)² - 2(0) + 3 = 3
   • f(3) = (3)² - 2(3) + 3 = 9 - 6 + 3 = 6
3. Compare y-values: The y-values are 2 (vertex), 3 (at x = 0), and 6 (at x = 3).
4. Determine highest and lowest y-values: The lowest y-value is 2, and the highest y-value is 6.
5. Write the range: The range is [2, 6].

In this example, even though the parabola opens upwards, the restricted domain limits the maximum y-value to 6.

Example with a Restricted Domain: g(x) = -x² + 4x - 1, for 1 ≤ x ≤ 4

1. Find the vertex:
   • h = -b / 2a = -4 / (2 * -1) = 2
   • k = g(2) = -(2)² + 4(2) - 1 = -4 + 8 - 1 = 3 The vertex is (2, 3).
2. Evaluate the function at the endpoints:
   • g(1) = -(1)² + 4(1) - 1 = -1 + 4 - 1 = 2
   • g(4) = -(4)² + 4(4) - 1 = -16 + 16 - 1 = -1
3. Compare y-values: The y-values are 3 (vertex), 2 (at x = 1), and -1 (at x = 4).
4. Determine highest and lowest y-values: The lowest y-value is -1, and the highest y-value is 3.
5. Write the range: The range is [-1, 3].

Here, the parabola opens downwards, and the restricted domain includes the vertex. The minimum y-value is -1, which occurs at x = 4.

Real-World Applications

Understanding the range of a quadratic function is useful in various real-world applications.

• Projectile Motion: In physics, the height of a projectile (e.g., a ball thrown into the air) can often be modeled by a quadratic function. The range of this function would tell you the possible heights the projectile can reach. The maximum height would be the upper bound of the range.
• Optimization Problems: In business and economics, quadratic functions are used to model cost, revenue, and profit. Finding the range can help determine the possible profit levels. For example, if a profit function is quadratic and opens downwards, the maximum profit is the upper bound of the range.
• Engineering: Quadratic functions can describe the shape of suspension cables in bridges or the trajectory of a robot arm. The range can help determine the spatial limits of these structures or movements.

Example: Projectile Motion

A ball is thrown upwards from an initial height of 1 meter with an initial velocity of 20 m/s. The height h(t) of the ball at time t (in seconds) is given by:

h(t) = -5t² + 20t + 1

Find the maximum height the ball reaches.

1. Identify coefficients: a = -5, b = 20, c = 1.
2. Direction of parabola: Since a = -5 < 0, the parabola opens downwards.
3. t-coordinate of vertex: h = -b / 2a = -20 / (2 * -5) = -20 / -10 = 2.
4. h-coordinate of vertex: k = h(2) = -5(2)² + 20(2) + 1 = -20 + 40 + 1 = 21.
5. Range: Since the parabola opens downwards and the vertex is at (2, 21), the range is (-∞, 21].

The maximum height the ball reaches is 21 meters.

Common Mistakes to Avoid

• Forgetting to check the sign of a: This determines whether the parabola opens upwards or downwards, which is crucial for determining the range.
• Incorrectly calculating the vertex: Double-check your calculations for h and k. A small error can lead to a completely incorrect range.
• Ignoring domain restrictions: If the domain is restricted, you must evaluate the function at the endpoints of the domain and consider those values when determining the range.
• Confusing range with domain: The range refers to the set of all possible y-values, while the domain refers to the set of all possible x-values.
• Assuming the range is always all real numbers: For quadratic functions, the range is always bounded either from below (if a > 0) or from above (if a < 0).

Conclusion

Finding the range of a quadratic function is a straightforward process once you understand the key concepts and steps involved. By identifying the coefficients, determining the direction of the parabola, finding the vertex, and considering any domain restrictions, you can accurately determine the range. Mastering these techniques will not only enhance your understanding of quadratic functions but also enable you to solve real-world problems involving quadratic models. Remember to practice with various examples to solidify your understanding and avoid common mistakes.