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

Summary

This 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).