Math Snap

STEP 1

What is this asking?
How does changing $y = x^3$ to $y = -2x^3 - 7$ affect the graph?
Watch out!
Don't mix up vertical stretches/shrinks with shifts up/down!
Also, be crystal clear about which axis the reflection is happening over.

STEP 2

1. Analyze the Transformation
2. Interpret the Changes

STEP 3

Alright, let's break down this funky function transformation!
We're starting with our basic cubic function, $y = x^3$, which we all know and love.
It's getting some serious upgrades!

STEP 4

First up, we've got that negative sign in front of the $2x^3$.
This bad boy reflects our graph over the x-axis.
Think of it like flipping a pancake – everything that was up is now down, and vice versa!

STEP 5

Next, that 2 in front of the $x^3$ is a vertical stretch.
It's like grabbing the graph and pulling it vertically away from the x-axis, making it twice as tall at every point.

STEP 6

Lastly, we've got that $-7$ hanging out at the end.
This is a vertical shift, moving our graph down by 7 units.
Imagine the whole graph just sliding down 7 spots on the y-axis.

STEP 7

So, putting it all together, we've got a reflection over the x-axis, a vertical stretch by a factor of 2, and a vertical shift down by 7 units.

The graph will be reflected over the x-axis, shifted down 7 units and vertical stretch of 2.