Rewrite The Quadratic Function In Standard Form

Rewriting Quadratic Functions in Standard Form: A Comprehensive Guide

The quadratic function, a cornerstone of algebra, finds applications across various fields, from physics to finance. Understanding how to manipulate and rewrite quadratic functions is crucial for solving problems and interpreting their graphical representations. This comprehensive guide delves into the intricacies of rewriting quadratic functions in standard form, exploring various methods and providing ample examples to solidify your understanding.

What is Standard Form?

Before diving into the rewriting process, it's essential to define the standard form of a quadratic function. A quadratic function in standard form is expressed as:

f(x) = ax² + bx + c

where:

• a, b, and c are constants (real numbers).
• a ≠ 0 (if a were 0, the function would be linear, not quadratic).
• x is the independent variable.

The standard form provides valuable insights into the quadratic function's characteristics, including its parabola's shape, vertex, and intercepts. The coefficient 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0), while 'c' represents the y-intercept (the point where the parabola intersects the y-axis).

Methods for Rewriting Quadratic Functions in Standard Form

Several methods can be employed to rewrite quadratic functions into standard form. The most common methods involve expanding expressions, completing the square, and using vertex form. Let's explore each method in detail.

1. Expanding and Simplifying

This method is straightforward when the quadratic function is given in a factored or expanded but unsimplified form. It involves multiplying out any brackets and then combining like terms to obtain the standard form.

Example 1:

Rewrite the function f(x) = (x + 2)(x - 3) in standard form.

Solution:

1. Expand the brackets: f(x) = x(x - 3) + 2(x - 3)
2. Simplify: f(x) = x² - 3x + 2x - 6
3. Combine like terms: f(x) = x² - x - 6

Therefore, the standard form of the function is f(x) = x² - x - 6.

Example 2:

Rewrite the function f(x) = 2(x + 1)² - 5 in standard form.

Solution:

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

The standard form of the function is f(x) = 2x² + 4x - 3.

This method is particularly useful when the quadratic is initially presented in a factored or partially expanded format. Its simplicity makes it an ideal starting point for many quadratic manipulations.

2. Completing the Square

Completing the square is a powerful technique for rewriting quadratic functions, especially when you need to find the vertex of the parabola represented by the function. This method involves manipulating the quadratic expression to create a perfect square trinomial.

Example 3:

Rewrite the function f(x) = x² + 6x + 5 in standard form using completing the square.

Solution:

1. Group the x terms: f(x) = (x² + 6x) + 5
2. Find half of the coefficient of x (6/2 = 3) and square it (3² = 9). Add and subtract this value inside the parenthesis: f(x) = (x² + 6x + 9 - 9) + 5
3. Factor the perfect square trinomial: f(x) = (x + 3)² - 9 + 5
4. Simplify: f(x) = (x + 3)² - 4

While this isn't in standard form ax²+bx+c, it's equally valid and provides the vertex form, from which the standard form is easily derived by expanding the squared term. Expanding this gives us: f(x) = x² + 6x + 9 - 4 = x² + 6x + 5. This confirms our process.

Example 4:

Rewrite the function f(x) = 2x² - 8x + 3 using completing the square.

Solution:

1. Factor out the coefficient of x² from the x terms: f(x) = 2(x² - 4x) + 3
2. Find half of the coefficient of x (-4/2 = -2) and square it ((-2)² = 4). Add and subtract this value inside the parenthesis: f(x) = 2(x² - 4x + 4 - 4) + 3
3. Factor the perfect square trinomial: f(x) = 2((x - 2)² - 4) + 3
4. Distribute the 2: f(x) = 2(x - 2)² - 8 + 3
5. Simplify: f(x) = 2(x - 2)² - 5

Expanding this vertex form will again confirm it simplifies back to the original quadratic function. This demonstrates the power of completing the square in transforming between forms.

Completing the square is particularly useful for identifying the vertex of the parabola, which is (-h, k) in the vertex form a(x-h)² + k. This is not immediately apparent in the standard form.

3. Using Vertex Form

The vertex form of a quadratic function is given by:

f(x) = a(x - h)² + k

where (h, k) represents the coordinates of the vertex of the parabola. If you know the vertex and one other point on the parabola, you can use this form to determine the quadratic function, which can then be expanded to standard form.

Example 5:

A parabola has a vertex at (2, -1) and passes through the point (0, 3). Find the quadratic function in standard form.

Solution:

1. Substitute the vertex coordinates into the vertex form: f(x) = a(x - 2)² - 1
2. Substitute the coordinates of the other point (0, 3): 3 = a(0 - 2)² - 1
3. Solve for 'a': 3 = 4a - 1 => 4a = 4 => a = 1
4. Substitute 'a' back into the vertex form: f(x) = (x - 2)² - 1
5. Expand to obtain standard form: f(x) = x² - 4x + 4 - 1 = x² - 4x + 3

Therefore, the quadratic function in standard form is f(x) = x² - 4x + 3.

This method is efficient when the vertex and an additional point are known. It leverages the geometrical properties of the parabola to simplify the process of finding the quadratic function.

Applications and Significance of Standard Form

Rewriting quadratic functions in standard form is not merely an algebraic exercise; it holds significant practical implications:

• Graphing: The standard form readily provides the y-intercept (c). The vertex can be found using the formula x = -b/2a, which allows for accurate sketching of the parabola.

• Solving Quadratic Equations: The standard form is crucial for applying the quadratic formula, factoring, or other methods to solve quadratic equations (finding the x-intercepts or roots of the quadratic function).

• Optimization Problems: In optimization problems (finding maximum or minimum values), the standard form and the vertex are essential for identifying the optimal value of the function.

• Modeling Real-World Phenomena: Quadratic functions model various real-world scenarios, including projectile motion, area calculations, and profit maximization. Standard form helps analyze these models effectively.

Conclusion

Rewriting quadratic functions in standard form is a fundamental skill in algebra with far-reaching applications. The methods discussed—expanding and simplifying, completing the square, and using vertex form—provide versatile tools for manipulating quadratic functions and extracting valuable information about their behavior. Mastering these techniques equips you to solve a wider range of problems and effectively analyze quadratic relationships in diverse contexts. Remember to always check your work by expanding your final answer to ensure it matches the original function. Practice is key to developing fluency and confidence in rewriting quadratic functions in standard form.