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### Course: Get ready for Precalculus>Unit 3

Lesson 3: Scaling functions

# Scaling functions horizontally: examples

The function f(k⋅x) is a horizontal scaling of f. See multiple examples of how we relate the two functions and their graphs, and determine the value of k.

## Want to join the conversation?

• Transformations of functions is the most trickier and interesting topic I've seen since joining khan academy. Scaling vertically and horizontally have connection, don't they ? if we scale by the same factor, are they the same in the linear function y=x and different in y=x^2 • Its just so round about, it does not make too much sense.
I am going to attempt to explain this just so I can better understand it.

g(x) = f(2x)
means: in this case the input of g is twice as big as input of f or f input has to be multiplied by two for it to equal g input. This is the same as g(x/2) = f(x) (we have to divide input of g by two just for it to equal input of f)

In summary, if we want to graph g, we should explain what g does to make more intuitive sense.
Since g(x/2) = f(x), then g does the following.
g(takes in input of f and divide it by two)

so
g(f(-4/2)) = (x = -2, y = is the same)
g(f(4/2)) = (x = 2, y = is the same)

since we are only changing x (input) and not touching the f(x) (output) we just keep the output the same for g(x) and f(x). • Does anyone understand how he got the x values for the table? Why are the 1/2x values the one used in the graph? Why aren't those just the x values? • Explanation 1:
Look carefully at both function f(x) and g(x).

You will see that the only difference between them is that in g(x), x is multiplied by (1/2).

So, g(x) = f(x/2).

Explanation 2:
f(x) = (x - 4)^2 - 4 ---- {Given in the question).
Define a new variable b. Let b = (x/2).
Substitute b into function f.
f(b) = (b - 4)^2 - 4
Since b = x/2.
f((x/2)) = ((x/2) - 4)^2 - 4
This is the exact same function as g(x).

----------------------------------------

He used those x values because they have an easy to determine y value.

Hope this helps.
• Where did he get g(x)= f(0.5x)? It looks like he just ignored the whole rest of the equation. • He got g(x) = f(0.5x) from the first function of the graph of f(x). If you look at both of the equations of f(x) and g(x) you will notice that they both have the same horizontal translation and vertical translation than that of the parent function of x^2. The only change is that g(x) is a horizontal stretch by a factor of 2 than f(x). Thus he ignored the rest part of the equation since that was not required for graphing. If by any chance the graph of g(x) was to be graphed on the basic of the parent function then, yes, all of the characteristics of the graph needs to be in mind. But in this example you are graphing g(x) on the basis of f(x) so doing the translation which has been done already will lead to incorrect results. That is why he graphed g(x) = f(0.5x) rather than graphing the whole equation again.

Hope this helps
• if g(x) is half of f(x), then wouldn't the graph move half of x to the left (B/c all the x values are half less) • why is it 3x and not x/3 for g(x)? • It’s like I’m thinking of this concept in the opposite way can someone pls break this down Barney style • For the second question, why am i not able to solve the question by just subbing in x values and plotting the graph? • When I tried doing this problem on my own I came to f(x)=g(1/3x) but Sal did f(3x)=g(x) are they the same thing?
(1 vote) • Why are horizontal transformations counterintuitive?
(1 vote) • They're not. Think of it this way:

Say you have the curve y=x³-3. To shift the curve down 5, we replace y by y+5. This is because if the point (x, y) is on the original curve, we want (x, y-5) to satisfy the new equation. So we increase y by 5 so that when (x, y-5) is plugged in, the 5s cancel and we get the original, true expression back.

So, (2, 5) lies on the curve y=x³-3. This means (2, 0) will lie on the down-shifted curve. The right-hand side still evaluates to 2³-3=5, not 0, so to edit the equation, we subtract 5 from y to rebalance. (2, 0) does satisfy y-5=x³-3.

That's how it works in any direction. To shift down 2, replace y by y+2.
To shift up 2, replace y by y-2.
To shift left 2, replace x by x+2.
To shift right 2, replace x by x-2.

All of the shifts are in the opposite direction that you might expect because we are working to cancel out the shift to keep the equation true.
(1 vote)