Math / Algebra

QuestionThe cubic function $y = x^3$ is transformed into $y = -2x^3 - 7$ . What effect will this have on the graph of the function?
The graph will be reflected over the x-axis, shifted down 7 units and vertical stretch of 2. The graph will be reflected over the x-axis, shifted up 7 units and vertical stretch of 2. The graph will be reflected over the y-axis, shifted down 2 units and vertical shrink of 2. The graph will be reflected over the y-axis, shifted up 2 units and vertical shrink of 2.

Studdy Solution

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.

STEP 8

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

Was this helpful?

Studdy Solution

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.

STEP 8

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

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? How does changing y=x3y = x^3y=x3 to y=−2x3−7y = -2x^3 - 7y=−2x3−7 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 Transformation2. Interpret the Changes

STEP 3

Alright, let's break down this funky function transformation!We're starting with our basic cubic function, y=x3y = x^3y=x3, 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 2x32x^32x3.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 x3x^3x3 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 −7-7−7 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**.

STEP 8

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

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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.

1. Analyze the Transformation
2. Interpret the Changes

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!

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!

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.

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.

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.