Quadratic Function Whose Zeros Are And

Let's explore how to construct a quadratic function when its zeros are known. Understanding the relationship between zeros and the function itself unlocks a powerful tool in algebra, allowing us to model various real-world scenarios with precision.

Understanding Quadratic Functions and Their Zeros

A quadratic function is a polynomial function 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 opens either upwards or downwards depending on the sign of a.

The zeros of a quadratic function, also known as roots or x-intercepts, are the values of x for which f(x) = 0. These are the points where the parabola intersects the x-axis. A quadratic function can have two distinct real zeros, one real zero (a repeated root), or no real zeros (two complex roots).

The Significance of Zeros

Zeros provide crucial information about the quadratic function:

• Factoring: Knowing the zeros allows us to factor the quadratic expression.
• Graphing: Zeros are key points for sketching the parabola.
• Problem Solving: Zeros often represent solutions to real-world problems modeled by quadratic functions.

Constructing a Quadratic Function from Given Zeros

The fundamental principle behind constructing a quadratic function from its zeros lies in the factor theorem. This theorem states that if r is a zero of a polynomial function f(x), then (x - r) is a factor of f(x).

Therefore, if we know the zeros of a quadratic function, say r₁ and r₂, we can express the function in factored form as:

f(x) = a(x - r₁)(x - r₂)

where a is a non-zero constant. This constant determines the vertical stretch or compression of the parabola.

Step-by-Step Guide

Let's break down the process into clear, manageable steps:

1. Identify the Zeros: Determine the values of r₁ and r₂.
2. Form the Factors: Create the factors (x - r₁) and (x - r₂).
3. Multiply the Factors: Multiply the factors together to obtain a quadratic expression.
4. Introduce the Constant 'a': Multiply the quadratic expression by a constant a.
5. Specify the Value of 'a' (If Possible): If additional information is provided (e.g., a point on the parabola), use it to determine the value of a. Otherwise, leave it as a variable.
6. Express the Function: Write the final quadratic function in the form f(x) = a(x - r₁)(x - r₂) or f(x) = ax² + bx + c.

Illustrative Examples

Let's solidify our understanding with several examples.

Example 1:

Construct a quadratic function whose zeros are 3 and -2.

1. Identify the Zeros: r₁ = 3, r₂ = -2
2. Form the Factors: (x - 3) and (x - (-2)) = (x + 2)
3. Multiply the Factors: (x - 3)(x + 2) = x² + 2x - 3x - 6 = x² - x - 6
4. Introduce the Constant 'a': f(x) = a(x² - x - 6)
5. Specify the Value of 'a': Assume a = 1 (unless additional information is given).
6. Express the Function: f(x) = x² - x - 6

Therefore, one quadratic function with zeros 3 and -2 is f(x) = x² - x - 6. Note that f(x) = 2(x² - x - 6) or f(x) = -1(x² - x - 6) would also have the same zeros.

Example 2:

Construct a quadratic function whose zeros are -1 and 5 and passes through the point (2, -9).

1. Identify the Zeros: r₁ = -1, r₂ = 5

2. Form the Factors: (x - (-1)) = (x + 1) and (x - 5)

3. Multiply the Factors: (x + 1)(x - 5) = x² - 5x + x - 5 = x² - 4x - 5

4. Introduce the Constant 'a': f(x) = a(x² - 4x - 5)

5. Specify the Value of 'a': Since the function passes through (2, -9), we have:

   -9 = a(2² - 4(2) - 5) -9 = a(4 - 8 - 5) -9 = a(-9) a = 1

6. Express the Function: f(x) = 1(x² - 4x - 5) = x² - 4x - 5

Therefore, the quadratic function is f(x) = x² - 4x - 5.

Example 3:

Construct a quadratic function that has a double root at x = 4.

1. Identify the Zeros: r₁ = 4, r₂ = 4 (double root)
2. Form the Factors: (x - 4) and (x - 4)
3. Multiply the Factors: (x - 4)(x - 4) = x² - 8x + 16
4. Introduce the Constant 'a': f(x) = a(x² - 8x + 16)
5. Specify the Value of 'a': Assume a = 1 (unless additional information is given).
6. Express the Function: f(x) = x² - 8x + 16

Therefore, one quadratic function with a double root at x = 4 is f(x) = x² - 8x + 16.

The Vertex Form and Its Relationship to Zeros

While the factored form f(x) = a(x - r₁)(x - r₂) is excellent for identifying zeros, the vertex form provides direct information about the vertex of the parabola. The vertex form is given by:

f(x) = a(x - h)² + k

where (h, k) are the coordinates of the vertex.

Converting from Factored Form to Vertex Form

If you have the zeros of a quadratic function and want to find its vertex form, you can follow these steps:

1. Find the Zeros: Determine r₁ and r₂.
2. Find the x-coordinate of the Vertex (h): The x-coordinate of the vertex is the average of the zeros: h = (r₁ + r₂) / 2.
3. Find the y-coordinate of the Vertex (k): Substitute h into the factored form of the function to find k: k = f(h) = a(h - r₁)(h - r₂). If you have already determined 'a' using another point, use that value. Otherwise, remember that 'a' will remain a variable.
4. Write the Vertex Form: Substitute h and k into the vertex form: f(x) = a(x - h)² + k.

Example:

Convert the quadratic function f(x) = x² - x - 6 (from Example 1) to vertex form.

1. Find the Zeros: From Example 1, the zeros are 3 and -2.
2. Find the x-coordinate of the Vertex (h): h = (3 + (-2)) / 2 = 1/2
3. Find the y-coordinate of the Vertex (k): k = f(1/2) = (1/2)² - (1/2) - 6 = 1/4 - 1/2 - 6 = -25/4
4. Write the Vertex Form: f(x) = (x - 1/2)² - 25/4

Therefore, the vertex form of the quadratic function is f(x) = (x - 1/2)² - 25/4. The vertex is at (1/2, -25/4).

Applications of Constructing Quadratic Functions from Zeros

The ability to construct a quadratic function from its zeros has numerous applications in various fields.

• Physics: Modeling projectile motion. The zeros represent the points where the projectile hits the ground.
• Engineering: Designing parabolic arches and bridges.
• Economics: Modeling profit and cost functions. The zeros can represent break-even points.
• Computer Graphics: Creating curves and surfaces.
• Optimization Problems: Finding maximum or minimum values of quadratic expressions.

Example Application: Projectile Motion

A ball is thrown into the air. It lands 8 meters away. The maximum height it reaches is 4 meters. Assuming a parabolic trajectory, determine the quadratic function that models the ball's path.

1. Establish a Coordinate System: Let the starting point of the ball be (0, 0). Then the landing point is (8, 0).

2. Identify the Zeros: r₁ = 0, r₂ = 8

3. Form the Factors: x and (x - 8)

4. Multiply the Factors: x(x - 8) = x² - 8x

5. Introduce the Constant 'a': f(x) = a(x² - 8x)

6. Use the Vertex Information: The maximum height occurs at the vertex. Since the parabola is symmetric, the x-coordinate of the vertex is the midpoint of the zeros: h = (0 + 8) / 2 = 4. The y-coordinate of the vertex is the maximum height, so k = 4. Therefore, the vertex is (4, 4). Substitute this into f(x) = a(x² - 8x):

   4 = a(4² - 8(4)) 4 = a(16 - 32) 4 = a(-16) a = -1/4

7. Express the Function: f(x) = (-1/4)(x² - 8x) = (-1/4)x² + 2x

Therefore, the quadratic function that models the ball's path is f(x) = (-1/4)x² + 2x.

Common Mistakes to Avoid

• Forgetting the Constant 'a': Always remember to include the constant a when constructing the quadratic function. This constant allows for vertical stretching or compression and is crucial for accurately modeling the function.
• Incorrectly Forming Factors: Ensure you correctly form the factors as (x - r₁) and (x - r₂). Pay close attention to the signs.
• Algebraic Errors: Be careful when multiplying the factors and simplifying the expression. Double-check your work to avoid errors.
• Misinterpreting Double Roots: Understand that a double root means the parabola touches the x-axis at only one point (the vertex lies on the x-axis).
• Incorrectly Applying Additional Information: If given a point on the parabola, substitute its coordinates correctly to solve for a.

Advanced Concepts and Extensions

• Complex Zeros: Quadratic functions can have complex zeros. If a quadratic function has complex zeros, they will always occur in conjugate pairs.
• Relationship to the Quadratic Formula: The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, is used to find the zeros of a quadratic function in the standard form.
• Vieta's Formulas: Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation ax² + bx + c = 0 with roots r₁ and r₂, Vieta's formulas state that r₁ + r₂ = -b/a and r₁r₂ = c/a.
• Polynomial Functions of Higher Degree: The concept of constructing a function from its zeros extends to polynomial functions of higher degree. If you know all the zeros of a polynomial function, you can express it in factored form.

Conclusion

Constructing a quadratic function from its zeros is a fundamental skill in algebra with wide-ranging applications. By understanding the relationship between zeros, factors, and the constant 'a', we can accurately model various real-world scenarios. Mastering this skill provides a solid foundation for further exploration of polynomial functions and their applications in mathematics, science, and engineering. Remember to pay attention to detail, avoid common mistakes, and practice regularly to strengthen your understanding. With consistent effort, you'll confidently navigate the world of quadratic functions and their zeros.