TRANSFORMING QUADRATIC FUNCTIONS FROM GENERAL FORM TO STANDARD/VERTEX FORM AND VICE VERSA

WOW MATH
28 Oct 202017:26

Summary

TLDRThis video tutorial explains how to transform a quadratic function from its general form, y = ax^2 + bx + c, to its vertex or standard form, y = a(x-h)^2 + k, and vice versa. The speaker covers completing the square to form perfect square trinomials and converting equations step-by-step. Additionally, there are examples with solutions to solidify understanding, including how to expand functions back to general form. The tutorial ends with a short quiz to review key concepts. Ideal for students learning quadratic functions, it combines explanation, examples, and practice.

Takeaways

  • 🧮 The general form of a quadratic function is y = ax² + bx + c, and the vertex (or standard) form is y = a(x - h)² + k.
  • 🔄 To transform from general form to vertex form, group terms, complete the square, and adjust constants accordingly.
  • ✍️ Completing the square involves adding (b/2)² and subtracting it to form a perfect square trinomial.
  • 📐 Express the perfect square trinomial as the square of a binomial, which helps rewrite the equation in vertex form.
  • ✔️ The values of a, h, and k in the vertex form represent the quadratic's shape and position on the graph, with h and k marking the vertex.
  • 🔢 A perfect square trinomial example: x² - 8x + 16 can be factored as (x - 4)².
  • 📝 In another example, x² + 3x + 9/4 is expressed as (x + 3/2)² after completing the square.
  • 🔍 Factoring out the coefficient of x² is necessary when a ≠ 1 before completing the square, such as in the example y = 2x² + 6x - 5.
  • 🧑‍🏫 Expanding vertex form back into general form requires applying the distributive property and combining like terms.
  • ✅ Several examples are given to reinforce understanding of converting between general and vertex forms of quadratic functions, with step-by-step calculations.

Q & A

  • What is the general form of a quadratic function?

    -The general form of a quadratic function is y = ax^2 + bx + c.

  • What is the vertex form (standard form) of a quadratic function?

    -The vertex form (or standard form) of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

  • What is the first step in transforming a quadratic function from general form to vertex form?

    -The first step is to group the terms that contain x and factor out 'a' from those terms if a ≠ 1.

  • How do you complete the square when transforming a quadratic function to vertex form?

    -To complete the square, add (b/2)^2 to both sides of the equation inside the parentheses and subtract a(b/2)^2 from the constant term.

  • What is the purpose of completing the square in the context of transforming a quadratic function?

    -Completing the square allows us to express the quadratic function as a perfect square trinomial, which can be factored into the form (x - h)^2.

  • How can you find the values of 'h' and 'k' when converting to vertex form?

    -The value of 'h' is found by dividing the coefficient of x (b) by 2, and squaring it to get the perfect square. The value of 'k' is derived by simplifying the equation after completing the square.

  • How do you go from vertex form back to general form?

    -To revert from vertex form to general form, expand the binomial (x - h)^2, distribute the 'a' term, and combine like terms with any constant.

  • In the example y = x^2 - 6x + 14, what are the values of a, h, and k in vertex form?

    -For the equation y = x^2 - 6x + 14, the values are a = 1, h = 3, and k = 5 in vertex form.

  • What is the method to factor a perfect square trinomial into a binomial square?

    -To factor a perfect square trinomial, express it as (x - b/2)^2, where b is the coefficient of x in the original trinomial.

  • What are the steps to transform y = 2x^2 + 6x - 5 into vertex form?

    -First, factor out the 'a' value from the terms containing x (if a ≠ 1), complete the square by adding and subtracting (b/2)^2 inside the equation, then simplify the constant term to get the final vertex form.

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