Answered

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Use the equation [tex]y = \sqrt[3]{27x - 54} + 5[/tex].

Which is an equivalent equation of the form [tex]y = a \sqrt[3]{x - h} + k[/tex]?

A. [tex]y = -27 \sqrt[3]{x + 2} + 5[/tex]
B. [tex]x = -3 \sqrt[3]{x + 2} + 5[/tex]
C. [tex]y = 3 \sqrt[3]{x - 2} + 5[/tex]
D. [tex]y = 27 \sqrt[3]{x - 2} + 5[/tex]


Sagot :

Let's start with the given equation:
[tex]\[ y = \sqrt[3]{27x - 54} + 5 \][/tex]

We aim to rewrite it in the form:
[tex]\[ y = a \sqrt[3]{x - h} + k \][/tex]

Step-by-step, let's transform the given equation to match this form.

1. Identify and factor the expression inside the cube root:
[tex]\[ \sqrt[3]{27x - 54} \][/tex]

Notice that [tex]\(27x - 54\)[/tex] can be factored as:
[tex]\[ 27(x - 2) \][/tex]

2. Rewrite the equation with this factor:
[tex]\[ y = \sqrt[3]{27(x - 2)} + 5 \][/tex]

3. Extract the constant factor from inside the cube root:
Since [tex]\(\sqrt[3]{27} = 3\)[/tex]:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]

Now, we have matched the form:
[tex]\[ y = a \sqrt[3]{x - h} + k \][/tex]

where:
[tex]\[ a = 3, \, h = 2, \, k = 5 \][/tex]

Hence, the equivalent equation is:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]

So, the correct choice from the given options is:
[tex]\[ y = 3 \sqrt[3]{x - 2} + 5 \][/tex]
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