What are the characteristics of the graph of a quadratic function?

Decoding the Parabola: What Makes a Quadratic Function Graph Tick?

Ever stared at a U-shaped curve and wondered what secrets it held? That, my friends, is a parabola – the graph of a quadratic function. But it’s not just any curve; it’s got personality, defined by some key characteristics. Let’s break it down, shall we?

Cracking the Code of Quadratic Functions

So, what exactly is a quadratic function? Simply put, it’s a polynomial where the highest power of ‘x’ is two. Think of it like this: it’s the VIP of second-degree polynomials! The most common way to write it is:

f(x) = ax² + bx + c

Here, a, b, and c are just regular numbers, but ‘a’ can’t be zero – otherwise, it wouldn’t be a quadratic, would it? Now, that little ‘a’ value is super important. It tells us whether our parabola is smiling or frowning. If ‘a’ is positive, it’s a smile (opens upwards); if it’s negative, it’s a frown (opens downwards). Easy peasy!

But wait, there’s more! We can also write quadratic functions in vertex form:

f(x) = a(x – h)² + k

This form is like a cheat sheet because (h, k) is the vertex – the very tip or bottom of the U. Trust me, this makes graphing a whole lot easier.

And finally, we have the intercept form:

f(x) = a(x – p)(x – q)

Here, p and q are where the parabola crosses the x-axis. It’s like finding the parabola’s landing spots!

Unmasking the Parabola’s Secrets

Quadratic Forms: Choose Your Weapon

• Standard Form: f(x) = ax² + bx + c. Great for spotting the y-intercept and using the quadratic formula when things get tricky.
• Vertex Form: f(x) = a(x – h)² + k. The go-to for instantly knowing the vertex and understanding how the basic parabola y = x² has been shifted around.
• Intercept Form: f(x) = a(x – p)(x – q). Perfect for quickly identifying the x-intercepts.

Spotting Characteristics: A Detective’s Guide

• From the Graph: Just look! The vertex is the highest or lowest point, the axis of symmetry cuts it in half, and the intercepts are where it crosses the axes.
• From Standard Form: Use h = -b/2a to find the x-coordinate of the vertex, then plug that into the function to find the y-coordinate. The y-intercept is ‘c’.
• From Vertex Form: The vertex is right there in the equation: (h, k).
• From Intercept Form: The x-intercepts are p and q. The vertex is right in the middle of them.

So, there you have it! The parabola, demystified. Once you know these key characteristics, you can unlock the secrets of any quadratic function. Happy graphing!