How do you stretch or compress a function?

Stretching and Compressing Functions: Making Math Bend to Your Will

Ever wonder how mathematicians tweak and transform those curvy lines and shapes we call functions? Well, two of the coolest tricks in the book are stretching and compressing them. Think of it like playing with Silly Putty – you can make your function taller, wider, skinnier, or shorter, all while keeping its essential character intact. Let’s dive into how it’s done!

Vertical Stretches and Compressions: Up and Down We Go!

Vertical stretches and compressions are all about messing with the y-values, the output of your function. Imagine grabbing the graph and pulling it upwards (stretch) or squishing it downwards (compression). The x-values, the input, stay put.

Vertical Stretch: Reaching for the Sky

A vertical stretch happens when you multiply your whole function, f(x), by a number bigger than 1. We’re talking g(x) = a*f(x), where a is a big deal. This makes the graph reach higher, like it’s trying to touch the sky. Every y-value gets multiplied, plain and simple.

Example: Take the humble f(x) = x². Now, let’s say we want to stretch it vertically by a factor of 2. Boom! We get g(x) = 2x². Suddenly, all those y-values are twice as big, making the parabola skinnier and taller.

Vertical Compression: Squishing Time!

On the flip side, a vertical compression is when you multiply f(x) by a number between 0 and 1. So, g(x) = a*f(x), but this time a is a fraction. This squishes the graph down towards the x-axis, making it shorter and wider.

Example: Back to our f(x) = x². If we compress it vertically by 0.5, we get g(x) = 0.5x². Now the y-values are half their original size, flattening out the parabola.

Key Things to Remember About Vertical Moves:

• The points where the graph crosses the x-axis? They don’t budge.
• If you use a negative number for a, you also flip the graph upside down. Bonus move!

Horizontal Stretches and Compressions: Side to Side Action!

Now, let’s talk about messing with the x-values, the input. Horizontal stretches and compressions change the width of the graph, like squeezing it from the sides or pulling it outwards. The y-values stay put this time.

Horizontal Compression: Squeeze It In!

To compress a graph horizontally, you replace x with bx inside the function, where b is bigger than 1. So, g(x) = f(bx). This squeezes the graph towards the y-axis, making it narrower. It’s like you’re fast-forwarding the function.

Example: Let’s compress f(x) = x² horizontally by a factor of 2. We get g(x) = (2x)². The graph gets squeezed in, becoming skinnier than before.

Horizontal Stretch: Stretch It Out!

To stretch a graph horizontally, you also replace x with bx, but this time b is between 0 and 1. Again, g(x) = f(bx). This stretches the graph away from the y-axis, making it wider. Think of it as putting the function in slow motion.

Example: If we stretch f(x) = x² horizontally by a factor of 0.5, we get g(x) = (0.5x)². Now the graph gets pulled outwards, becoming wider.

Key Things to Remember About Horizontal Moves:

• The point where the graph crosses the y-axis? It stays put.
• The amount you stretch or compress is actually the opposite of what b looks like it should do. Tricky, right?
• If b is negative, you flip the graph left-to-right, like looking in a mirror.

Combining It All: The Transformation Tango

The real fun begins when you start combining these moves with shifts and flips. It’s like a mathematical dance! Just remember, the order matters. A good rule of thumb is to handle the horizontal stuff before you tackle the vertical stuff.

Example: Check out k(x) = f(½x + 1) – 3. This one’s got a horizontal stretch (by 2, thanks to the ½), a shift to the left, and a drop downwards. It’s a whole party of transformations!

Wrapping Up: Bend That Function to Your Will!

Stretching and compressing functions might sound complicated, but they’re really just about reshaping graphs to fit your needs. Once you get the hang of vertical and horizontal moves, you’ll be able to manipulate functions like a mathematical maestro. So go ahead, play around, and see what amazing shapes you can create!