Math  /  Algebra

QuestionWhat kind of transformation converts the graph of f(x)=7x10f(x)=-7 x-10 into the graph of g(x)=g(x)= x10x-10 ? horizontal stretch horizontal shrink vertical shrink vertical stretch

Studdy Solution

STEP 1

What is this asking? We're figuring out how to stretch or shrink f(x)f(x) to make it look like g(x)g(x). Watch out! Don't mix up horizontal and vertical transformations!
Also, remember a stretch makes things bigger and a shrink makes things smaller.

STEP 2

1. Compare the functions
2. Determine the transformation

STEP 3

Alright, let's **look** at our functions!
We've got f(x)=7x10f(x) = -7x - 10 and g(x)=x10g(x) = x - 10.
Notice that the constant term, 10-10, is the same in both.
So, we only need to worry about how to turn 7x-7x into xx.

STEP 4

Let's think about what happens when we multiply f(x)f(x) by a certain factor.
Let's call this factor cc.
So, we have cf(x)=c(7x10)c \cdot f(x) = c \cdot (-7x - 10).

STEP 5

We want cf(x)c \cdot f(x) to equal g(x)g(x).
Focus on the xx terms.
We want c(7x)c \cdot (-7x) to equal xx.

STEP 6

To find cc, we can set up the equation c(7x)=xc \cdot (-7x) = x.
Now, let's **solve** for cc!

STEP 7

To isolate cc, we'll divide both sides of the equation by 7x-7x.
This gives us c=x7x. c = \frac{x}{-7x}.

STEP 8

We can simplify this fraction by dividing both the numerator and the denominator by xx.
Remember that xx divided by xx equals 11, so we get c=17=17. c = \frac{1}{-7} = -\frac{1}{7}.

STEP 9

So, we found that c=17c = -\frac{1}{7}.
This means we multiply f(x)f(x) by 17-\frac{1}{7} to get g(x)g(x).
Let's check: 17f(x)=17(7x10)=17(7x)17(10)=x+107. -\frac{1}{7} \cdot f(x) = -\frac{1}{7}(-7x - 10) = -\frac{1}{7} \cdot (-7x) - \frac{1}{7} \cdot (-10) = x + \frac{10}{7}. Oops, that's not quite right!
We got the xx part right, but the constant term is off.
The problem only asked about stretches and shrinks, so we're only looking at the coefficient of xx.

STEP 10

Since we're multiplying the *function* f(x)f(x) by 17-\frac{1}{7}, this is a *vertical* transformation.
Because 17\frac{1}{7} is between -1 and 1, it's a *shrink*.
And because it's negative, it's also a reflection across the x-axis.
The question didn't ask about reflections, so we'll ignore that.

STEP 11

The transformation is a vertical shrink.

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