Math  /  Algebra

QuestionWhat kind of transformation converts the graph of f(x)=6x1f(x)=6 x-1 into the graph of g(x)=xg(x)=x- 1 ? horizontal shrink vertical shrink horizontal stretch vertical stretch

Studdy Solution

STEP 1

What is this asking? How do we change the graph of f(x)=6x1f(x) = 6x - 1 to get the graph of g(x)=x1g(x) = x - 1? Watch out! Don't mix up horizontal and vertical stretches and shrinks!
A *horizontal* change affects the input (xx) and a *vertical* change affects the output (f(x)f(x)).

STEP 2

1. Rewrite the function
2. Identify the transformation

STEP 3

Let's **rewrite** g(x)g(x) so it looks more like f(x)f(x).
We want to see what happened to that 66 in front of the xx.
Notice that xx is the same as 166x\frac{1}{6} \cdot 6x, right?
Multiplying by 16\frac{1}{6} and then by 66 gets us back where we started because 166=1\frac{1}{6} \cdot 6 = 1.
So, we can **rewrite** g(x)g(x) as: g(x)=x1=166x1g(x) = x - 1 = \frac{1}{6} \cdot 6x - 1

STEP 4

Now, let's **rewrite** this in terms of f(x)f(x).
Remember, f(x)=6x1f(x) = 6x - 1.
So, we can **substitute** f(x)f(x) into our expression for g(x)g(x): g(x)=166x1=16f(x)+160=16f(x)g(x) = \frac{1}{6} \cdot 6x - 1 = \frac{1}{6} \cdot f(x) + \frac{1}{6} \cdot 0 = \frac{1}{6} f(x) We added zero in the form of 160\frac{1}{6} \cdot 0 to make the substitution clearer.

STEP 5

We found that g(x)=16f(x)g(x) = \frac{1}{6} f(x).
This means the output of gg is 16\frac{1}{6} times the output of ff.
Since the output, or the *vertical* component, is being changed, we know this is a *vertical* transformation.

STEP 6

Since we're multiplying f(x)f(x) by a number between 0 and 1 (that's our 16\frac{1}{6}), this is a **vertical shrink**!
If we were multiplying by a number *greater* than 1, it would be a vertical stretch.

STEP 7

The transformation that converts f(x)f(x) into g(x)g(x) is a **vertical shrink**.

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