Finding a Quadratic Function Given X-Intercepts and a Point

Introduction

Key Concepts

1. Quadratic Functions: Definition and Standard Form

A quadratic function is a second-degree polynomial function of the form:

where:

• a ≠ 0: Determines the parabola's opening direction and its width.
• b and c: Affect the position and shape of the parabola.

The graph of a quadratic function is a parabola that either opens upwards (if a > 0) or downwards (if a < 0).

2. X-Intercepts of a Quadratic Function

The x-intercepts (also known as roots or zeros) of a quadratic function are the points where the graph intersects the x-axis. These points satisfy the equation f(x) = 0. If a quadratic function has two distinct x-intercepts, they can be denoted as (p, 0) and (q, 0), where p and q are the roots of the equation.

3. Factored Form of a Quadratic Function

When the x-intercepts are known, the quadratic function can be expressed in its factored form:

Here, p and q represent the x-intercepts, and a is a leading coefficient that affects the parabola's width and direction.

4. Converting Factored Form to Standard Form

To convert the factored form to standard form, expand the product:

This allows for easy identification of the coefficients a, b, and c in the standard form.

5. Determining the Leading Coefficient (a)

The leading coefficient a can be determined using an additional point (x₁, y₁) that lies on the parabola. Substituting x₁ and y₁ into the factored form allows for solving the value of a:

6. Example Problem

Find the quadratic function given the x-intercepts (2, 0) and (5, 0), and the point (3, 4).

1. Write the factored form: $$ f(x) = a(x - 2)(x - 5) $$
2. Use the point (3, 4) to find a: $$ 4 = a(3 - 2)(3 - 5) \\ 4 = a(1)(-2) \\ 4 = -2a \\ a = -2 $$
3. Write the quadratic function: $$ f(x) = -2(x - 2)(x - 5) $$
4. Convert to standard form: $$ f(x) = -2(x^2 - 7x + 10) \\ f(x) = -2x^2 + 14x - 20 $$

7. The Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. The vertex (h, k) can be found using the formula:

Using the earlier example:

8. Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images. It is given by:

In the example, the axis of symmetry is x = 3.5.

9. Graphing the Quadratic Function

To graph the quadratic function:

• Plot the x-intercepts (2, 0) and (5, 0).
• Determine and plot the vertex (3.5, 4.5).
• Draw the axis of symmetry x = 3.5.
• Sketch the parabola opening downwards (since a = -2 < 0).
• Plot additional points if necessary for accuracy.

10. Application in Real-World Problems

Quadratic functions are widely used to model real-world scenarios such as projectile motion, optimizing areas and profits, and calculating trajectories. Understanding how to derive these functions from given data allows for precise modeling and problem-solving in various disciplines.

Advanced Concepts

1. Derivation Using the Vertex Form

While the factored form is useful when x-intercepts are known, the vertex form provides another pathway to derive the quadratic function, especially when the vertex and a point are known:

However, when x-intercepts and an additional point are given, one can still derive the vertex form by first finding the standard form and then converting it.

2. Completing the Square

Completing the square is a method used to convert a quadratic function from standard form to vertex form. Given a standard quadratic equation:

We can complete the square as follows:

1. Factor out a from the first two terms: $$ f(x) = a\left(x^2 + \frac{b}{a}x\right) + c $$
2. Add and subtract \(\left(\frac{b}{2a}\right)^2\) inside the parentheses: $$ f(x) = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c $$
3. Simplify to the vertex form: $$ f(x) = a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c \\ f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c \\ f(x) = a\left(x - h\right)^2 + k $$

   where \( h = -\frac{b}{2a} \) and \( k = c - \frac{b^2}{4a} \).

3. Discriminant and Nature of Roots

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

The discriminant indicates the nature of the roots:

• If D > 0: Two distinct real roots (two x-intercepts).
• If D = 0: One real root (vertex touches the x-axis).
• If D < 0: No real roots (no x-intercepts).

In the context of finding a quadratic function with given x-intercepts and a point, the discriminant confirms the presence of those intercepts.

4. Systems of Equations Approach

Another method to find the quadratic function is by setting up a system of equations using the standard form and the given points:

Given x-intercepts (p, 0) and (q, 0), and a point (x₁, y₁), the quadratic function can be expressed as:

Substituting the x-intercepts:

Substituting the additional point:

Solving the system of equations (1), (2), and (3) simultaneously will yield the values of a, b, and c.

5. Application of System of Equations: Example

Find the quadratic function with x-intercepts at (-1, 0) and (3, 0), and passing through the point (1, 8).

1. Set up the standard form: $$ f(x) = ax^2 + bx + c $$
2. Substitute the x-intercepts:
   • (-1, 0): $$ 0 = a(-1)^2 + b(-1) + c \\ 0 = a - b + c \quad \text{(1)} $$
   • (3, 0): $$ 0 = a(3)^2 + b(3) + c \\ 0 = 9a + 3b + c \quad \text{(2)} $$
3. Substitute the additional point (1, 8): $$ 8 = a(1)^2 + b(1) + c \\ 8 = a + b + c \quad \text{(3)} $$
4. Solve the system of equations:
   • From (1) and (3): $$ 0 = a - b + c \quad \text{(1)} \\ 8 = a + b + c \quad \text{(3)} \\ \text{Subtract (1) from (3):} \\ 8 = 2b \\ b = 4 $$
   • Substitute b = 4 into (1): $$ 0 = a - 4 + c \\ a + c = 4 \quad \text{(4)} $$
   • Substitute b = 4 into (2): $$ 0 = 9a + 12 + c \\ 9a + c = -12 \quad \text{(5)} $$
   • Subtract (4) from (5): $$ 9a + c - (a + c) = -12 - 4 \\ 8a = -16 \\ a = -2 $$
   • Substitute a = -2 into (4): $$ -2 + c = 4 \\ c = 6 $$
   • Quadratic function: $$ f(x) = -2x^2 + 4x + 6 $$

This approach systematically finds the coefficients by leveraging the given points to form equations.

6. Interdisciplinary Connections

Quadratic functions are integral in various fields beyond pure mathematics:

• Physics: Modeling projectile motion, where the path of an object under gravity follows a parabolic trajectory.
• Economics: Determining profit maximization and cost minimization scenarios, where revenue and cost functions are often quadratic.
• Engineering: Designing structures and analyzing stresses that follow quadratic relationships.
• Biology: Population models where growth rates may follow quadratic trends under certain conditions.

Understanding quadratic functions thus provides a foundational tool applicable across diverse scientific and practical contexts.

7. Optimization Problems

Finding the maximum or minimum value of a quadratic function is crucial in optimization problems:

• Maximum: If the parabola opens downwards (a < 0), the vertex represents the maximum point.
• Minimum: If the parabola opens upwards (a > 0), the vertex represents the minimum point.

For example, determining the optimal price point to maximize revenue involves finding the vertex of the corresponding quadratic revenue function.

8. Complex Numbers and Quadratic Equations

When the discriminant D < 0, the quadratic equation has complex roots. While in the context of finding a quadratic function with real x-intercepts, complex roots are not applicable, understanding them is essential in advanced studies:

• Imaginary Roots: Expressed as a ± bi, where i is the imaginary unit.
• Conjugate Pairs: Complex roots occur in conjugate pairs for real-coefficient polynomials.

This knowledge paves the way for exploring polynomial functions beyond real numbers.

9. Transformations of Quadratic Functions

Understanding how different transformations affect the graph of a quadratic function enhances the ability to graph and interpret these functions:

• Vertical Shifts: Adding or subtracting a constant shifts the parabola up or down.
• Horizontal Shifts: Changing the input variable shifts the parabola left or right.
• Reflections: Multiplying by -1 reflects the parabola over the x-axis.
• Scaling: Changing the value of a affects the width and direction of the parabola.

For instance, the function \( f(x) = a(x - h)^2 + k \) represents a parabola with vertex (h, k), shifted horizontally and vertically.

10. Systems of Quadratic Functions

Analyzing systems involving multiple quadratic functions can lead to finding points of intersection, optimization, and other complex problem-solving scenarios. This involves solving simultaneous quadratic equations, which may require methods like substitution, elimination, or using the quadratic formula.

Example:

Find the points of intersection between \( f(x) = x^2 - 4x + 3 \) and \( g(x) = -x^2 + 6x - 5 \).

1. Set f(x) = g(x): $$ x^2 - 4x + 3 = -x^2 + 6x - 5 $$
2. Combine like terms: $$ 2x^2 - 10x + 8 = 0 \\ x^2 - 5x + 4 = 0 $$
3. Factor the quadratic equation: $$ (x - 1)(x - 4) = 0 \\ x = 1 \text{ or } x = 4 $$
4. Find corresponding y-values:
   • For x = 1: $$ f(1) = 1 - 4 + 3 = 0 \\ \text{Point: } (1, 0) $$
   • For x = 4: $$ f(4) = 16 - 16 + 3 = 3 \\ \text{Point: } (4, 3) $$

The functions intersect at (1, 0) and (4, 3).

Comparison Table

Summary and Key Takeaways