Math / Algebra

Question11. The function $g(x)=4(x-6)^{3}-2$ was created by transforming $f(x)=x^{3}$ . To create $g(x), f(x)$ was -Choose the correct answer - - , then was $\qquad$ and then was -Choose the correct answer - . - Choose the correct answer - stretched vertically by a factor of 4 Clear All compressed vertically by a factor 4

Studdy Solution

STEP 1

What is this asking? How do we transform the function $f(x) = x^3$ into $g(x) = 4(x-6)^3 - 2$ ? Watch out! Remember the order of transformations matters!

STEP 2

1. Horizontal Shift
2. Vertical Stretch
3. Vertical Shift

STEP 3

We start with $f(x) = x^3$ .
To get the $(x-6)$ part in $g(x)$ , we need to shift $f(x)$ horizontally to the right by 6 units.

STEP 4

This gives us a new function, let's call it $h(x) = (x-6)^3$ .
Remember, shifting to the right means subtracting from $x$ .

STEP 5

Now, we need that 4 in front.
This means we vertically stretch $h(x)$ by a factor of 4.

STEP 6

Multiplying the function by 4 gives us another new function, let's call it $j(x) = 4(x-6)^3$ .

STEP 7

Finally, we have that $-2$ at the end.
This means we shift $j(x)$ downwards vertically by 2 units.

STEP 8

Subtracting 2 gives us our final function $g(x) = 4(x-6)^3 - 2$ .

STEP 9

$f(x)$ was shifted to the right by 6, then stretched vertically by a factor of 4, and then shifted down by 2.

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Studdy Solution

STEP 1

What is this asking? How do we transform the function $f(x) = x^3$ into $g(x) = 4(x-6)^3 - 2$ ? Watch out! Remember the order of transformations matters!

STEP 2

1. Horizontal Shift
2. Vertical Stretch
3. Vertical Shift

STEP 3

We start with $f(x) = x^3$ .
To get the $(x-6)$ part in $g(x)$ , we need to shift $f(x)$ horizontally to the right by 6 units.

STEP 4

This gives us a new function, let's call it $h(x) = (x-6)^3$ .
Remember, shifting to the right means subtracting from $x$ .

STEP 5

Now, we need that 4 in front.
This means we vertically stretch $h(x)$ by a factor of 4.

STEP 6

Multiplying the function by 4 gives us another new function, let's call it $j(x) = 4(x-6)^3$ .

STEP 7

Finally, we have that $-2$ at the end.
This means we shift $j(x)$ downwards vertically by 2 units.

STEP 8

Subtracting 2 gives us our final function $g(x) = 4(x-6)^3 - 2$ .

STEP 9

$f(x)$ was shifted to the right by 6, then stretched vertically by a factor of 4, and then shifted down by 2.

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Studdy Solution

STEP 1

What is this asking? How do we transform the function f(x)=x3f(x) = x^3f(x)=x3 into g(x)=4(x−6)3−2g(x) = 4(x-6)^3 - 2g(x)=4(x−6)3−2? Watch out! Remember the order of transformations matters!

STEP 2

1. Horizontal Shift2. Vertical Stretch3. Vertical Shift

STEP 3

We **start** with f(x)=x3f(x) = x^3f(x)=x3.To get the (x−6)(x-6)(x−6) part in g(x)g(x)g(x), we need to **shift** f(x)f(x)f(x) **horizontally** to the **right** by **6 units**.

STEP 4

This gives us a **new function**, let's call it h(x)=(x−6)3h(x) = (x-6)^3h(x)=(x−6)3.Remember, shifting to the *right* means *subtracting* from xxx.

STEP 5

Now, we need that **4** in front.This means we **vertically stretch** h(x)h(x)h(x) by a **factor of 4**.

STEP 6

**Multiplying** the function by **4** gives us another **new function**, let's call it j(x)=4(x−6)3j(x) = 4(x-6)^3j(x)=4(x−6)3.

STEP 7

Finally, we have that −2-2−2 at the end.This means we **shift** j(x)j(x)j(x) **downwards vertically** by **2 units**.

STEP 8

**Subtracting 2** gives us our **final function** g(x)=4(x−6)3−2g(x) = 4(x-6)^3 - 2g(x)=4(x−6)3−2.

STEP 9

f(x)f(x)f(x) was shifted to the **right** by **6**, then **stretched vertically** by a factor of **4**, and then shifted **down** by **2**.

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What is this asking? How do we transform the function $f(x) = x^3$ into $g(x) = 4(x-6)^3 - 2$ ? Watch out! Remember the order of transformations matters!

1. Horizontal Shift
2. Vertical Stretch
3. Vertical Shift

We start with $f(x) = x^3$ .
To get the $(x-6)$ part in $g(x)$ , we need to shift $f(x)$ horizontally to the right by 6 units.

This gives us a new function, let's call it $h(x) = (x-6)^3$ .
Remember, shifting to the right means subtracting from $x$ .

Now, we need that 4 in front.
This means we vertically stretch $h(x)$ by a factor of 4.

Multiplying the function by 4 gives us another new function, let's call it $j(x) = 4(x-6)^3$ .

Finally, we have that $-2$ at the end.
This means we shift $j(x)$ downwards vertically by 2 units.

Subtracting 2 gives us our final function $g(x) = 4(x-6)^3 - 2$ .

$f(x)$ was shifted to the right by 6, then stretched vertically by a factor of 4, and then shifted down by 2.