Math  /  Algebra

QuestionFor each of the following, write a formula for the function, gg, obtained when the graph of ff is transformed as described. a. The graph of f(x)=x3f(x)=\sqrt[3]{x} is vertically stretched by a factor of 3 , then shifted to the right 4 units and down 6 units. g(x)=g(x)= \square b. The graph of f(x)=xf(x)=\sqrt{x} is horizontally compressed by a factor of 14\frac{1}{4}, then shifted to the left 7 units and down 3 units. g(x)=g(x)= \square
For additional help with this problem type, access the following resources: - TEXT Read College Algebra with Corequisite Support 2e 3.5 Transformation of Functions of the text.

Studdy Solution

STEP 1

What is this asking? We need to find the new function, g(x)g(x), after stretching/compressing and shifting the graphs of f(x)=x3f(x) = \sqrt[3]{x} and f(x)=xf(x) = \sqrt{x}. Watch out! Remember the order of transformations matters!
Also, horizontal transformations can be tricky – they often do the opposite of what you might expect!

STEP 2

1. Vertically Stretched Cube Root
2. Horizontally Compressed Square Root

STEP 3

Alright, let's **tackle the first transformation**: a vertical stretch by a factor of 3\mathbf{3}.
This means we **multiply** the *entire* function f(x)f(x) by 3\mathbf{3}.
So, we have 3f(x)=3x33 \cdot f(x) = 3\sqrt[3]{x}.
Why? Because a vertical stretch affects the *output* of the function, making it 3\mathbf{3} times larger!

STEP 4

Next, we **shift to the right** by 4\mathbf{4} units.
Remember, horizontal shifts work inside the function, and they're a little sneaky.
Shifting to the *right* means we **subtract** 4\mathbf{4} from xx.
So, we now have 3x433\sqrt[3]{x - 4}.
Think of it like this: to get the same output, we need a *larger* input, hence the subtraction.

STEP 5

Finally, we **shift down** by 6\mathbf{6} units.
This affects the output of the function, so we subtract 6\mathbf{6} from the *entire* expression.
This gives us our final function: g(x)=3x436g(x) = 3\sqrt[3]{x - 4} - 6.
Down means subtracting, simple as that!

STEP 6

Let's start with the **horizontal compression** by a factor of 14\frac{\mathbf{1}}{\mathbf{4}}.
This means we **multiply** xx *inside* the function by 4\mathbf{4}.
Remember, horizontal transformations are tricky!
Compressing by 14\frac{1}{4} actually means multiplying by the reciprocal, which is 4\mathbf{4}.
So we have 4x\sqrt{4x}.

STEP 7

Now, we **shift to the left** by 7\mathbf{7} units.
Since we're moving left, we **add** 7\mathbf{7} to xx *inside* the function.
This gives us 4(x+7)\sqrt{4(x + 7)}.
Adding 7\mathbf{7} means we need a *smaller* input to get the same output, hence the addition.

STEP 8

Lastly, we **shift down** by 3\mathbf{3} units.
We **subtract** 3\mathbf{3} from the *entire* expression, giving us our final function: g(x)=4(x+7)3g(x) = \sqrt{4(x + 7)} - 3.

STEP 9

a. g(x)=3x436g(x) = 3\sqrt[3]{x - 4} - 6 b. g(x)=4(x+7)3g(x) = \sqrt{4(x + 7)} - 3

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