Write A Quadratic Function F Whose Zeros Are And

Let's explore the process of constructing a quadratic function, denoted as f, when its zeros (or roots) are known. This process leverages the fundamental relationship between the zeros of a quadratic function and its algebraic representation. We'll delve into the underlying principles, step-by-step methods, and illustrative examples to provide a comprehensive understanding.

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 zeros of a quadratic function are the values of x for which f(x) = 0. These zeros represent the x-intercepts of the parabola defined by the quadratic function when graphed on the Cartesian plane.

The zeros of a quadratic function can be real or complex numbers. If the discriminant (b² - 4ac) is positive, the function has two distinct real roots. If the discriminant is zero, the function has one real root (a repeated root). If the discriminant is negative, the function has two complex conjugate roots.

The Relationship Between Zeros and Factors

The cornerstone of constructing a quadratic function from its zeros lies in the relationship between the zeros and the factors of the quadratic expression. If r₁ and r₂ are the zeros of a quadratic function f(x), then (x - r₁) and (x - r₂) are factors of f(x). This is a direct consequence of the Factor Theorem.

Therefore, we can express the quadratic function as:

f(x) = a(x - r₁)(x - r₂)

where a is a non-zero constant. The value of a determines the vertical stretch or compression of the parabola and whether it opens upwards (a > 0) or downwards (a < 0).

Steps to Construct a Quadratic Function from its Zeros

Let's outline the step-by-step process to construct a quadratic function f(x) given its zeros r₁ and r₂:

1. Identify the Zeros: Determine the given zeros of the quadratic function, r₁ and r₂.

2. Form the Factors: Construct the factors (x - r₁) and (x - r₂) corresponding to the zeros.

3. Write the General Form: Express the quadratic function in the general form f(x) = a(x - r₁)(x - r₂), where a is a constant.

4. Determine the Value of 'a':

   • If additional information is provided, such as a point on the parabola (other than the zeros), substitute the coordinates of the point into the general form and solve for a.
   • If no additional information is provided, you can choose any non-zero value for a. Often, a = 1 is chosen for simplicity, resulting in the simplest quadratic function with the given zeros.

5. Expand and Simplify (Optional): Expand the product (x - r₁)(x - r₂) and multiply by a to obtain the quadratic function in the standard form f(x) = ax² + bx + c. This step is not always necessary but can be useful for certain applications.

Example 1: Constructing a Quadratic Function with Zeros 2 and -3

Let's construct a quadratic function f(x) with zeros r₁ = 2 and r₂ = -3.

1. Identify the Zeros: r₁ = 2 and r₂ = -3.

2. Form the Factors: (x - r₁) = (x - 2) and (x - r₂) = (x - (-3)) = (x + 3).

3. Write the General Form: f(x) = a(x - 2)(x + 3).

4. Determine the Value of 'a': Let's assume a = 1 for simplicity.

5. Expand and Simplify:

   • f(x) = 1(x - 2)(x + 3)
   • f(x) = (x² + 3x - 2x - 6)
   • f(x) = x² + x - 6

Therefore, the quadratic function with zeros 2 and -3 is f(x) = x² + x - 6.

Example 2: Constructing a Quadratic Function with Zeros 1 and 4 and Passing Through the Point (2, -2)

Let's construct a quadratic function f(x) with zeros r₁ = 1 and r₂ = 4 that passes through the point (2, -2).

1. Identify the Zeros: r₁ = 1 and r₂ = 4.

2. Form the Factors: (x - r₁) = (x - 1) and (x - r₂) = (x - 4).

3. Write the General Form: f(x) = a(x - 1)(x - 4).

4. Determine the Value of 'a': Since the function passes through (2, -2), we have f(2) = -2. Substitute x = 2 and f(x) = -2 into the general form:

   • -2 = a(2 - 1)(2 - 4)
   • -2 = a(1)(-2)
   • -2 = -2a
   • a = 1

5. Expand and Simplify:

   • f(x) = 1(x - 1)(x - 4)
   • f(x) = (x² - 4x - x + 4)
   • f(x) = x² - 5x + 4

Therefore, the quadratic function with zeros 1 and 4 and passing through the point (2, -2) is f(x) = x² - 5x + 4.

Example 3: Constructing a Quadratic Function with Complex Zeros 1 + i and 1 - i

Let's construct a quadratic function f(x) with complex zeros r₁ = 1 + i and r₂ = 1 - i.

1. Identify the Zeros: r₁ = 1 + i and r₂ = 1 - i.

2. Form the Factors:

   • (x - r₁) = (x - (1 + i)) = (x - 1 - i)
   • (x - r₂) = (x - (1 - i)) = (x - 1 + i)

3. Write the General Form: f(x) = a(x - 1 - i)(x - 1 + i).

4. Determine the Value of 'a': Let's assume a = 1 for simplicity.

5. Expand and Simplify:

   • f(x) = 1(x - 1 - i)(x - 1 + i)
   • f(x) = ((x - 1) - i)((x - 1) + i)
   • Using the difference of squares formula, (A - B)(A + B) = A² - B², where A = (x - 1) and B = i:
   • f(x) = (x - 1)² - i²
   • f(x) = (x² - 2x + 1) - (-1) (Since i² = -1)
   • f(x) = x² - 2x + 1 + 1
   • f(x) = x² - 2x + 2

Therefore, the quadratic function with complex zeros 1 + i and 1 - i is f(x) = x² - 2x + 2.

Dealing with Repeated Zeros

If a quadratic function has a repeated zero (i.e., a double root), then r₁ = r₂ = r. In this case, the quadratic function can be expressed as:

f(x) = a(x - r)²

This represents a parabola that touches the x-axis at x = r but does not cross it.

Example 4: Constructing a Quadratic Function with a Repeated Zero of 3

Let's construct a quadratic function f(x) with a repeated zero of r = 3.

1. Identify the Zero: r = 3.

2. Form the Factor: (x - r) = (x - 3).

3. Write the General Form: f(x) = a(x - 3)².

4. Determine the Value of 'a': Let's assume a = 1 for simplicity.

5. Expand and Simplify:

   • f(x) = 1(x - 3)²
   • f(x) = (x² - 6x + 9)

Therefore, the quadratic function with a repeated zero of 3 is f(x) = x² - 6x + 9.

The Role of the Leading Coefficient 'a'

The leading coefficient a in the quadratic function f(x) = a(x - r₁)(x - r₂) plays a crucial role in determining the shape and orientation of the parabola.

• Sign of 'a':
  • If a > 0, the parabola opens upwards (the vertex is the minimum point).
  • If a < 0, the parabola opens downwards (the vertex is the maximum point).
• Magnitude of 'a':
  • If |a| > 1, the parabola is vertically stretched (narrower).
  • If 0 < |a| < 1, the parabola is vertically compressed (wider).
  • If |a| = 1, the parabola has a standard width.

By adjusting the value of a, we can create a family of quadratic functions that all share the same zeros but have different shapes.

Using Vertex Form

Another useful form for representing a quadratic function is the vertex form:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola. If you know the vertex and one other point on the parabola, you can determine the value of a and write the equation in vertex form. While this method doesn't directly use the zeros, it provides an alternative approach when vertex information is available.

Applications

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

• Physics: Modeling projectile motion, where the zeros represent the points where the projectile hits the ground.
• Engineering: Designing parabolic reflectors for antennas and solar concentrators.
• Economics: Modeling cost and revenue functions.
• Computer Graphics: Creating curves and shapes.

Conclusion

Constructing a quadratic function from its zeros is a fundamental skill in algebra with wide-ranging applications. By understanding the relationship between the zeros and the factors of a quadratic expression, we can easily determine the equation of the function. The process involves identifying the zeros, forming the factors, writing the general form, and determining the value of the leading coefficient a. Whether dealing with real or complex zeros, repeated zeros, or additional information such as a point on the parabola, the principles remain the same. This comprehensive guide provides the knowledge and examples necessary to confidently construct quadratic functions from their zeros.