Lesson 2: Understanding Quadratic Functions and Parabolas
What is a Quadratic Function?
A quadratic function is a function of the form f(x) = ax2 + bx + c, where “a” cannot be 0. The cool thing about these functions is that their graphs form a shape called a parabola! Parabolas may open upward or downward. They have the “U” shape.
How to Graph a Parabola
The most basic parabola has an equation f(x) = x2. To plot a parabola, you can start by creating a table of values. Pick various x-values, plug them into the function, and then find the corresponding f(x)-values. Once you have a bunch of points, you can graph them on a coordinate plane and see the beautiful curve take shape!
What’s the Vertex?
The vertex is a significant part of the parabola—it’s the highest or lowest point, depending on the direction it opens. If you’re looking at a quadratic function in vertex form:
f(x) = a(x – h)2 + k Vertex (h, k)
You can easily spot the vertex at the point (h, k). This makes it super easy to graph the function!
Changing Standard Form to Vertex Form
Sometimes, we’ll need to convert a quadratic from its standard form to vertex form to find that helpful vertex. This is done through a process called “completing the square.” It’s like rearranging the equation a little bit—don’t worry; it’s easier than it sounds! You can add and subtract the same number to keep things balanced, which helps you identify the vertex quickly.
Completing the square can also help us solve quadratic equations to find their zeros, or x-intercepts. These are the points where the parabola crosses the x-axis, and they can be vital in figuring out where something (like a score in a game) becomes zero.
Real-Life Applications of Quadratics
You might be surprised at how often quadratic functions pop up in real life! For example, if you’ve ever played the “Angry Birds” game, you know that understanding how objects fly in a parabolic path is key to making it fun and realistic. The same goes for the path of a soccer ball when it’s kicked or a diver making a splash off a diving board. In basketball, players use quadratic functions to figure out the perfect angle and force to make a basket—pretty amazing, right?
Wrapping It Up
Quadratic functions are not just about numbers; they’re a fascinating part of the world around us! In the coming lessons, we’ll dive deeper into how to solve quadratic equations using methods like factoring and the Quadratic Formula. These tools will make it even easier to find those important x-intercepts. As we explore this topic together, keep in mind how these concepts connect to real-life scenarios, from sports to games and much more!
Grade 10 Parabola Quiz
