What Are General and Vertex Forms of Quadratic Functions?

TL;DR

The general form of a quadratic function is expressed as 'ax² + bx + c', affecting its shape and position, while the vertex form is '(x-h)² + k', which directly reveals the vertex's coordinates. Understanding these forms allows for easier graphing and manipulation of parabolas by highlighting how constants influence curvature and positioning.

Transcript

Hello. I'm Professor Von Schmohawk and welcome to Why U. In the previous lectures, we introduced "quadratic functions". In this lecture, we will examine two standard ways to write a quadratic function the general form and the vertex form and see how those forms are related to the function's graph. We have seen that any quadratic function of x can b... Read More

Key Insights

• 😃 The general form of a quadratic function is "ax^2 + bx + c" where a, b, and c determine the parabola's characteristics.
• 💱 Changing "a" in a quadratic function affects the curvature of the parabola.
• 💁 Vertex form of a quadratic function simplifies determining the vertex and sketching the parabola graph.
• 🤗 Positive and negative values of "a" in a quadratic function determine the direction in which the parabola opens.
• 💁 Constants h and k in the vertex form directly represent the horizontal and vertical coordinates of the parabola's vertex.
• 😃 Adjusting values of b and c in the general form shifts the parabola horizontally and vertically.
• 💁 Knowing the vertex coordinates in vertex form simplifies positioning the parabola on the Cartesian plane.

Questions & Answers

Q: How do the constants a, b, and c affect the shape and position of a quadratic function?

The constant a determines the curvature, b and c affect the position, with b shifting the parabola horizontally and c vertically.

Q: What is the significance of writing a quadratic function in vertex form?

Vertex form, with constants h and k as the vertex coordinates, simplifies determining and sketching the parabola's graph.

Q: What do positive and negative values of "a" signify in a quadratic function?

Positive a values make the parabola open upwards, while negative a values make it open downwards.

Q: How can the vertex form of a quadratic function help in determining the position of the parabola on a graph?

By knowing the values of h and k, which are the vertex coordinates, the parabola's position can be easily pinpointed on the graph.

Summary & Key Takeaways

• Quadratic functions can be written in general form as "ax^2 + bx + c" where a, b, and c determine the graph's shape and position.

• The constants a, b, and c impact the curvature, position, and vertex of the parabola.

• Quadratic functions can also be written in vertex form as "(x-h)^2 + k" where h and k are the coordinates of the vertex.