Algebra 67 - Deriving the Vertex Form of a Quadratic Function

TL;DR

This lecture explains the general form and vertex form of quadratic functions, highlighting their differences and how to calculate the vertex coordinates.

Transcript

Hello. I'm Professor Von Schmohawk and welcome to Why U. In the previous lecture, we examined two ways in which quadratic functions can be written. These two common forms of a quadratic function are the "general form" and the "vertex form". Regardless of its algebraic form the graph of a quadratic function in a single variable is always a parabola.... Read More

Key Insights

• 👔 Quadratic functions can be written in general form (ax^2 + bx + c) or vertex form (a(x-h)^2 + k).
• 💁 The constants in each form determine the shape, width, and position of the parabola.
• 💁 The vertex coordinates can be calculated using the constants in the general form or directly obtained from the h and k values in the vertex form.
• 👈 Shifting a quadratic function vertically is done by adding or subtracting a constant value (k), while shifting it horizontally is achieved by replacing x with (x-h).
• 💨 The vertex form provides a simpler way to identify the coordinates of the vertex.
• ☺️ The vertex form of a quadratic function can be derived from the general form by replacing x with (x-h) and adding k.
• 🤙 The vertex form can also be obtained through a process called completing the square.

Questions & Answers

Q: What are the two common forms of quadratic functions?

The two common forms of quadratic functions are the general form (ax^2 + bx + c) and the vertex form (a(x-h)^2 + k).

Q: How can the vertex coordinates be calculated for a quadratic function in general form?

The x-coordinate of the vertex can be calculated as -b/2a, and the vertical coordinate can be found by substituting the x-coordinate into the quadratic equation.

Q: What are the advantages of using the vertex form to represent a quadratic function?

The vertex form directly provides the horizontal (h) and vertical (k) coordinates of the vertex, making it easier to identify the position of the parabola.

Q: How can the graph of a quadratic function be shifted vertically or horizontally?

To shift the graph vertically, a constant value (k) is added or subtracted to the function. To shift it horizontally, x is replaced with (x-h), where h represents the number of units to shift the graph.

Summary & Key Takeaways

• Quadratic functions can be written in general form (ax^2 + bx + c) or vertex form (a(x-h)^2 + k).

• The constants in each form determine the shape, position, and width of the parabola.

• The vertex coordinates of a quadratic function can be calculated using the constants a, b, and c in the general form or by directly reading the h and k values in the vertex form.