Quadratic Function Whose Zeros Are And

The quadratic function, a cornerstone of algebra, opens doors to understanding various phenomena, from projectile motion to optimization problems. When you know the zeros (or roots) of a quadratic function, you possess a powerful key to unlocking its equation and properties.

Understanding Quadratic Functions and Zeros

A quadratic function is 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. Graphically, these are the points where the parabola intersects the x-axis.

• Why are zeros important? Zeros provide critical information about the function's behavior. They tell us where the function's output is zero, which can represent real-world solutions in various applications.
• Relationship to factors: Each zero corresponds to a factor of the quadratic expression. If x = r is a zero, then (x - r) is a factor.

Finding the Quadratic Function Given Its Zeros

Let's say we are given two zeros, x₁ and x₂. We can construct the quadratic function using the following steps:

1. Form the factors: Create the factors (x - x₁) and (x - x₂).
2. Multiply the factors: Multiply the factors to get the quadratic expression: (x - x₁)(x - x₂).
3. General form: The quadratic function can be written as f(x) = a(x - x₁)(x - x₂), where a is a constant. This constant determines the "stretch" or "compression" and the direction (upward or downward) of the parabola.

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

Let's work through a concrete example. Suppose we are given that the zeros of a quadratic function are x₁ = 2 and x₂ = -3.

1. Form the factors: The factors are (x - 2) and (x - (-3)) = (x + 3).
2. Multiply the factors: (x - 2)(x + 3) = x² + 3x - 2x - 6 = x² + x - 6
3. General form: The quadratic function can be written as f(x) = a(x² + x - 6).

Notice that we have a family of quadratic functions, all with the same zeros. The specific value of a will determine which function we have. For example:

• If a = 1, then f(x) = x² + x - 6.
• If a = 2, then f(x) = 2x² + 2x - 12.
• If a = -1, then f(x) = -x² - x + 6.

All these functions have zeros at x = 2 and x = -3.

Determining the Value of 'a'

Often, you'll be given additional information that allows you to determine the value of a. This usually comes in the form of a point that lies on the parabola, i.e., a specific (x, y) coordinate.

Example: Suppose we know the zeros are 2 and -3, and the function passes through the point (1, -8). We can use this information to find a.

1. Start with the general form: f(x) = a(x² + x - 6)
2. Substitute the point (1, -8): -8 = a(1² + 1 - 6)
3. Solve for a: -8 = a(1 + 1 - 6) -8 = a(-4) a = 2
4. Final function: Therefore, the quadratic function is f(x) = 2(x² + x - 6) = 2x² + 2x - 12.

The Vertex Form of a Quadratic Function

Another useful form of a quadratic function is the vertex form: f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex is the point where the parabola reaches its minimum (if a > 0) or maximum (if a < 0) value.

Converting from Standard Form to Vertex Form:

1. Complete the square: Take the quadratic function in standard form (ax² + bx + c) and complete the square. This involves manipulating the expression to create a perfect square trinomial.
2. Rewrite in vertex form: After completing the square, the expression will be in the form a(x - h)² + k.

Finding the Vertex from Zeros:

If you know the zeros of a quadratic function, you can find the x-coordinate of the vertex h by averaging the zeros: h = (x₁ + x₂)/2. Then, substitute this value of h back into the function to find the y-coordinate of the vertex k = f(h).

Example (Continuing from previous example): f(x) = 2x² + 2x - 12

1. Find the x-coordinate of the vertex: h = (2 + (-3))/2 = -1/2
2. Find the y-coordinate of the vertex: k = f(-1/2) = 2(-1/2)² + 2(-1/2) - 12 = 2(1/4) - 1 - 12 = 1/2 - 13 = -25/2
3. Vertex form: Therefore, the vertex form of the quadratic function is f(x) = 2(x + 1/2)² - 25/2.

Properties of Quadratic Functions

• Parabola: The graph of a quadratic function is a parabola.
• Axis of Symmetry: The parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex. The equation of the axis of symmetry is x = h.
• Direction: If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards.
• Vertex: The vertex represents the minimum or maximum point of the function.
• Discriminant: The discriminant, Δ = b² - 4ac, tells us about the nature of the roots:
  • Δ > 0: Two distinct real roots (the parabola intersects the x-axis at two points).
  • Δ = 0: One real root (the parabola touches the x-axis at one point – the vertex lies on the x-axis).
  • Δ < 0: No real roots (the parabola does not intersect the x-axis).

Applications of Quadratic Functions

Quadratic functions are used extensively in various fields:

• Physics: Modeling projectile motion (e.g., the trajectory of a ball thrown in the air).
• Engineering: Designing bridges, arches, and other structures.
• Economics: Optimizing profit and cost functions.
• Computer Graphics: Creating curves and surfaces.
• Optimization Problems: Finding the maximum or minimum values of a function subject to certain constraints.

Example: Projectile Motion

The height h(t) of a projectile launched vertically upwards can be modeled by a quadratic function of time t: h(t) = -1/2gt² + v₀t + h₀, where g is the acceleration due to gravity, v₀ is the initial velocity, and h₀ is the initial height. The zeros of this function represent the times when the projectile hits the ground. The vertex represents the maximum height reached by the projectile.

Working with Complex Zeros

While many quadratic functions have real zeros, some have complex zeros. Complex zeros always come in conjugate pairs. If a + bi is a zero of a quadratic function with real coefficients, then a - bi is also a zero.

Constructing a Quadratic Function with Complex Zeros:

The process is similar to that with real zeros.

1. Form the factors: If the zeros are a + bi and a - bi, the factors are (x - (a + bi)) and (x - (a - bi)).
2. Multiply the factors: Multiply the factors and simplify. The imaginary terms will cancel out, resulting in a quadratic expression with real coefficients.
3. General form: The quadratic function can be written as f(x) = a(quadratic expression), where a is a real constant.

Example: Suppose the zeros are 1 + i and 1 - i.

1. Form the factors: (x - (1 + i)) and (x - (1 - i))
2. Multiply the factors: (x - (1 + i))(x - (1 - i)) = (x - 1 - i)(x - 1 + i) = ((x - 1) - i)((x - 1) + i) = (x - 1)² - i² = x² - 2x + 1 - (-1) = x² - 2x + 2
3. General form: f(x) = a(x² - 2x + 2)

Techniques for Finding Zeros

Besides being given the zeros, here are the common techniques to find them:

• Factoring: If the quadratic expression can be factored easily, set each factor equal to zero and solve for x.

• Quadratic Formula: For any quadratic equation ax² + bx + c = 0, the quadratic formula gives the zeros:

  x = (-b ± √(b² - 4ac)) / 2a

• Completing the Square: Rewrite the quadratic expression in vertex form and then solve for x.

Tips and Tricks

• Check your work: Always substitute the zeros back into the quadratic function to verify that the result is zero.
• Pay attention to the sign of 'a': The sign of a determines whether the parabola opens upwards or downwards, which affects whether the vertex is a minimum or maximum.
• Use a graphing calculator: Graphing calculators can be helpful for visualizing quadratic functions and finding their zeros and vertex.
• Remember the relationship between zeros and factors: This is a fundamental concept for working with quadratic functions.

Advanced Considerations

• Multiplicity of Roots: A quadratic function can have a repeated root, meaning one zero appears twice. In this case, the parabola touches the x-axis at the vertex.
• Relationship to Higher-Degree Polynomials: The concepts of zeros and factors extend to polynomials of higher degrees. For example, a cubic function has three zeros (which may be real or complex, and some may be repeated).
• Vieta's Formulas: Vieta's formulas relate the coefficients of a polynomial to the sums and products of its roots. For a quadratic function ax² + bx + c = 0, Vieta's formulas state that the sum of the roots is -b/a and the product of the roots is c/a.

Common Mistakes to Avoid

• Forgetting the 'a' constant: Remember that there are infinitely many quadratic functions with the same zeros, differing only by a constant factor. Don't forget to find the value of a if you're given additional information.
• Incorrectly applying the quadratic formula: Be careful with the signs when using the quadratic formula.
• Making algebraic errors when multiplying factors: Double-check your work when multiplying the factors to avoid mistakes.
• Confusing zeros with the vertex: Zeros are the x-intercepts, while the vertex is the minimum or maximum point of the parabola.

Conclusion

Understanding how to construct a quadratic function from its zeros is a fundamental skill in algebra. By mastering the steps outlined above and practicing with various examples, you'll gain a deeper understanding of quadratic functions and their applications. From simple equations to complex real-world problems, the ability to work with quadratic functions and their zeros is a valuable asset in mathematics and beyond. Remember to focus on the relationship between zeros, factors, and the different forms of a quadratic function (standard form and vertex form). With practice, you'll become proficient in manipulating and analyzing quadratic functions with ease.