How To Write A Quadratic Function

Alright, let's dive into the world of quadratic functions! Whether you're tackling algebra homework, building simulations, or just curious about mathematical relationships, understanding how to write and manipulate quadratic functions is a valuable skill. This comprehensive guide will walk you through everything from the basic form to real-world applications.

Understanding the Basics of Quadratic Functions

A quadratic function is a polynomial function of degree two. This means the highest power of the variable (usually x) is 2. You'll recognize it by the x² term. The general form of a quadratic function is:

f(x) = ax² + bx + c

Where:

• f(x) represents the value of the function at x.
• x is the independent variable.
• a, b, and c are constants, with a not equal to 0 (otherwise, it would be a linear function).

Let's break down each component:

• a: This coefficient determines the shape and direction of the parabola. If a > 0, the parabola opens upwards (it's a "happy" parabola). If a < 0, the parabola opens downwards (a "sad" parabola). The absolute value of a dictates how "wide" or "narrow" the parabola is. Larger values of |a| result in a narrower parabola, while smaller values result in a wider one.

• b: This coefficient influences the position of the parabola's axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. Changing the value of b shifts the parabola horizontally.

• c: This constant represents the y-intercept of the parabola. It's the point where the parabola crosses the y-axis. In other words, f(0) = c.

Different Forms of Quadratic Functions

Besides the general form, quadratic functions can be expressed in other useful forms:

1. Standard Form (Vertex Form):

   f(x) = a(x - h)² + k

   This form is incredibly helpful because it directly reveals the vertex of the parabola, which is the point where the parabola reaches its minimum (if a > 0) or maximum (if a < 0) value. The vertex is located at the point (h, k). The axis of symmetry is the vertical line x = h. The a coefficient still determines the direction and "width" of the parabola.

2. Factored Form (Intercept Form):

   f(x) = a(x - r₁)(x - r₂)

   This form highlights the roots (or zeros) of the quadratic function, which are the x-values where the function crosses the x-axis (i.e., where f(x) = 0). The roots are r₁ and r₂. Again, a controls the direction and width. The x-intercepts are the points (r₁, 0) and (r₂, 0).

How to Write a Quadratic Function: Step-by-Step

Now, let's get practical. How do you actually write a quadratic function given specific information? Here's a breakdown of common scenarios:

1. Given the Vertex and a Point:

This is perfect for using the standard (vertex) form.

• Step 1: Start with the vertex form: f(x) = a(x - h)² + k

• Step 2: Substitute the vertex (h, k): You'll have f(x) = a(x - h)² + k with h and k filled in.

• Step 3: Use the given point (x, y) to find 'a': Substitute the x and y values of the given point into the equation. This leaves you with only 'a' as the unknown. Solve for 'a'.

• Step 4: Write the final equation: Substitute the values of a, h, and k back into the vertex form.

**Example:**  Find the equation of a parabola with vertex (2, -3) that passes through the point (4, 5).

*   *f(x) = a(x - 2)² - 3*
*   *5 = a(4 - 2)² - 3*
*   *5 = a(2)² - 3*
*   *5 = 4a - 3*
*   *8 = 4a*
*   *a = 2*

Therefore, the equation is *f(x) = 2(x - 2)² - 3*.  You could expand this to get the general form: *f(x) = 2x² - 8x + 5*.

2. Given the x-intercepts and a Point:

This calls for the factored (intercept) form.

• Step 1: Start with the factored form: f(x) = a(x - r₁)(x - r₂)

• Step 2: Substitute the x-intercepts (r₁ and r₂): Remember, the x-intercepts are the roots of the equation.

• Step 3: Use the given point (x, y) to find 'a': Substitute the x and y values of the given point into the equation and solve for 'a'.

• Step 4: Write the final equation: Substitute the values of a, r₁, and r₂ back into the factored form.

**Example:** Find the equation of a parabola with x-intercepts at -1 and 3 that passes through the point (1, 4).

*   *f(x) = a(x + 1)(x - 3)*
*   *4 = a(1 + 1)(1 - 3)*
*   *4 = a(2)(-2)*
*   *4 = -4a*
*   *a = -1*

Therefore, the equation is *f(x) = -1(x + 1)(x - 3)* or *f(x) = -(x + 1)(x - 3)*.  Expanding gives the general form: *f(x) = -x² + 2x + 3*.

3. Given Three Points:

This requires a bit more algebra, but it's very doable using the general form.

• Step 1: Start with the general form: f(x) = ax² + bx + c

• Step 2: Substitute each point (x, y) into the equation: This will give you three equations with three unknowns (a, b, and c).

• Step 3: Solve the system of equations: Use any method you prefer (substitution, elimination, matrices) to solve for a, b, and c.

• Step 4: Write the final equation: Substitute the values of a, b, and c back into the general form.

**Example:** Find the equation of a parabola that passes through the points (1, 2), (2, 5), and (-1, 8).

*   Substituting (1, 2):  *a(1)² + b(1) + c = 2  => a + b + c = 2*
*   Substituting (2, 5):  *a(2)² + b(2) + c = 5  => 4a + 2b + c = 5*
*   Substituting (-1, 8): *a(-1)² + b(-1) + c = 8 => a - b + c = 8*

Now, solve this system of equations.  Here's one way to do it using elimination:

*   Subtract the first equation from the second:  *(4a + 2b + c) - (a + b + c) = 5 - 2  => 3a + b = 3*
*   Subtract the first equation from the third:  *(a - b + c) - (a + b + c) = 8 - 2  => -2b = 6  => b = -3*

Substitute *b = -3* into *3a + b = 3*:  *3a - 3 = 3  => 3a = 6  => a = 2*

Substitute *a = 2* and *b = -3* into *a + b + c = 2*: *2 - 3 + c = 2  => c = 3*

Therefore, *a = 2*, *b = -3*, and *c = 3*.  The equation is *f(x) = 2x² - 3x + 3*.

4. Transforming Between Forms:

• General to Standard (Vertex): Complete the square. This is a crucial algebraic technique.

  Example: Convert f(x) = x² + 6x + 5 to vertex form.

  1. Group the x² and x terms: f(x) = (x² + 6x) + 5
  2. Take half of the coefficient of the x term (which is 6), square it (3² = 9), and add and subtract it inside the parentheses: f(x) = (x² + 6x + 9 - 9) + 5
  3. Rewrite the expression inside the parentheses as a squared term: f(x) = (x + 3)² - 9 + 5
  4. Simplify: f(x) = (x + 3)² - 4

  Now it's in vertex form: f(x) = (x - (-3))² + (-4). The vertex is (-3, -4).

• Standard (Vertex) to General: Expand the squared term and simplify.

  Example: Convert f(x) = 2(x - 1)² + 3 to general form.

  1. Expand the squared term: f(x) = 2(x² - 2x + 1) + 3
  2. Distribute the 2: f(x) = 2x² - 4x + 2 + 3
  3. Simplify: f(x) = 2x² - 4x + 5

• Factored to General: Multiply the factors and simplify.

  Example: Convert f(x) = (x - 2)(x + 4) to general form.

  1. Multiply the factors: f(x) = x² + 4x - 2x - 8
  2. Simplify: f(x) = x² + 2x - 8

• General to Factored: Factor the quadratic expression (if possible).

  Example: Convert f(x) = x² - 5x + 6 to factored form.

  1. Find two numbers that multiply to 6 and add to -5: -2 and -3
  2. Write the factored form: f(x) = (x - 2)(x - 3)

Important Considerations and Tips

• The Discriminant: The discriminant (Δ) of a quadratic equation ax² + bx + c = 0 is given by Δ = b² - 4ac. It tells you about the nature of the roots:

  • If Δ > 0: Two distinct real roots (the parabola intersects the x-axis at two points).
  • If Δ = 0: One real root (a repeated root; the parabola touches the x-axis at one point – the vertex lies on the x-axis).
  • If Δ < 0: No real roots (the parabola does not intersect the x-axis). The roots are complex.

• Completing the Square: Mastering completing the square is essential for converting between general and vertex forms and for solving quadratic equations.

• Factoring: Practice factoring quadratic expressions. It will make finding the x-intercepts much easier.

• Quadratic Formula: The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is a guaranteed way to find the roots of any quadratic equation, even if it's not easily factorable.

• Graphing Calculators and Software: Use graphing calculators or software like Desmos or GeoGebra to visualize quadratic functions and check your work. Seeing the graph can often help you understand the relationships between the equation and its properties.

Real-World Applications of Quadratic Functions

Quadratic functions aren't just abstract mathematical concepts; they appear everywhere in the real world:

• Projectile Motion: The path of a projectile (like a ball thrown in the air) can be modeled by a quadratic function. The height of the ball at any given time can be determined using a quadratic equation.

• Optimization Problems: Quadratic functions are used to find the maximum or minimum values in various situations. For example, determining the dimensions of a rectangular garden that will maximize the area given a fixed amount of fencing.

• Engineering: Designing arches, bridges, and other structures often involves quadratic equations.

• Business and Economics: Modeling cost, revenue, and profit functions. For instance, determining the price that maximizes profit.

• Physics: Describing the motion of objects under constant acceleration.

FAQ

• Q: What happens if 'a' is zero in the general form?

  • A: If a = 0, the function becomes f(x) = bx + c, which is a linear function, not a quadratic function.

• Q: Can a quadratic function have no real roots?

  • A: Yes, if the discriminant (b² - 4ac) is negative. In this case, the parabola does not intersect the x-axis.

• Q: How do I find the y-intercept of a quadratic function?

  • A: The y-intercept is the value of f(x) when x = 0. In the general form, f(x) = ax² + bx + c, the y-intercept is simply c.

• Q: Is it always possible to factor a quadratic expression?

  • A: No. If the discriminant is negative or a perfect square is difficult to find, it might not be easily factorable with real numbers. You can always use the quadratic formula to find the roots.

Conclusion

Writing quadratic functions is a fundamental skill with widespread applications. By understanding the different forms (general, standard, and factored) and practicing the techniques for converting between them, you'll be well-equipped to tackle a variety of problems. Remember to leverage the power of the discriminant, completing the square, and graphing tools to deepen your understanding. Don't be afraid to practice with different examples, and soon you'll be writing quadratic functions with confidence!

How do you plan to use your newfound knowledge of quadratic functions? Are there any specific applications that you find particularly interesting?