How To Find The Vertex of a Parabola - Standard Form, Factored & Vertex Form

The Organic Chemistry Tutor
9 Sept 201718:03

Summary

TLDRThis video teaches how to find the vertex of a parabola from quadratic equations in three different forms: standard form, vertex form, and factored form. It explains the use of formulas like -b/2a to find the vertex in standard form, and how to identify the vertex directly in vertex form. The video also demonstrates how to calculate the midpoint of x-intercepts in factored form and provides step-by-step examples for each method. It emphasizes practical techniques for understanding the process, making it accessible for learners to apply in various problems.

Takeaways

  • 📐 The standard form of a quadratic equation is y = ax^2 + bx + c.
  • 🔍 To find the vertex of a parabola in standard form, use the formula x = -b/(2a) to get the x-coordinate.
  • 📌 Once you have the x-coordinate of the vertex, substitute it back into the original equation to find the y-coordinate.
  • 📈 If the coefficient of x^2 is positive, the parabola opens upward.
  • 🔢 For example, in the equation y = x^2 - 4x + 3, a = 1, b = -4, and c = 3.
  • 🤔 The vertex form of a quadratic equation is y = a(x - h)^2 + k, where the vertex is at the point (h, k).
  • 🔄 To find the vertex from the vertex form, simply read off the values of h and k.
  • 📊 For a quadratic equation in factored form, find the x-intercepts to determine the x-coordinate of the vertex.
  • 🧮 The x-coordinate of the vertex is the midpoint of the x-intercepts.
  • 📑 Two methods were demonstrated for finding the vertex from a factored form: using the midpoint of x-intercepts or converting to standard form and applying the vertex formula.
  • 🎓 The video provided step-by-step examples for each form, reinforcing the learning with practice problems.

Q & A

  • What is the formula to find the x-coordinate of the vertex in the standard form of a quadratic equation?

    -The formula to find the x-coordinate of the vertex is x = -b / 2a, where 'b' is the coefficient of the x-term and 'a' is the coefficient of the x²-term.

  • Once you find the x-coordinate of the vertex, how do you find the y-coordinate?

    -After finding the x-coordinate, substitute this x-value back into the original quadratic equation to calculate the y-coordinate of the vertex.

  • What is the vertex of the quadratic equation y = x² - 4x + 3?

    -The vertex of the equation y = x² - 4x + 3 is (2, -1).

  • How do you know whether a parabola opens upwards or downwards based on the equation in standard form?

    -The direction of the parabola depends on the sign of the 'a' coefficient in the equation y = ax² + bx + c. If 'a' is positive, the parabola opens upward. If 'a' is negative, it opens downward.

  • What is the vertex form of a quadratic equation, and how is the vertex determined?

    -The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex. The value of 'h' is the opposite of the number inside the parentheses, and 'k' remains the same.

  • How can you find the vertex from a quadratic equation in factored form?

    -To find the vertex in factored form, first find the x-intercepts by setting each factor to zero. The x-coordinate of the vertex is the midpoint of the x-intercepts, and the y-coordinate is found by substituting this x-value back into the equation.

  • In the factored form method, how do you find the midpoint of the x-intercepts?

    -The midpoint of the x-intercepts is found by taking the average of the two intercepts, calculated as (x₁ + x₂) / 2.

  • What are the x and y coordinates of the vertex for the equation y = -2(x - 2)(x - 5)?

    -The x-coordinate of the vertex is 3.5, and the y-coordinate is 4.5. The vertex is (3.5, 4.5).

  • Why is it easier to use the x-intercepts to find the vertex when a quadratic equation is in factored form?

    -It is easier because you can quickly calculate the x-intercepts, find their midpoint, and then use this value to find the vertex without having to expand the equation into standard form.

  • What is the vertex of the equation y = 4x² - 12x?

    -The vertex of the equation y = 4x² - 12x is (3/2, -9).

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الوسوم ذات الصلة
Quadratic equationsVertex formFactored formParabola vertexMath tutorialAlgebra tipsStep-by-step mathMath problemsAlgebraic equationsMath examples
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