7-18 Describing Transformations Suppose the graph of f is...

7-18 Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$
\begin{equation}(a)y=f(x-5)+2 \quad \text { (b) } y=f(x+1)-1\end{equation}

Key Concepts

Vertical Translation

Vertical translation involves moving a graph up or down without altering its shape. This is accomplished by adding or subtracting a constant to the entire function. Adding a constant moves the graph upward, and subtracting a constant moves it downward.

Function Transformation

Function transformation refers to the process of altering the appearance or position of a graph through modifications made directly to its function rule. This includes translations, reflections, stretching, and compressing, and it preserves the overall shape of the graph while moving it to a new position or altering its orientation.

Horizontal Translation

Horizontal translation is the process of shifting a graph left or right. It is achieved by modifying the x-variable in the function. For example, replacing x with x - h in the function's formula shifts the graph to the right by h units, while replacing x with x + h shifts it to the left by h units.

Transcript

00:01 All right, so let's first look at part a and try to describe what the transformations are here.

00:08 So the first thing we're going to do is we're going to look at that x minus 5.

00:12 So to figure out what's happening there, you might have already heard that if it's inside the parentheses, minus means to the right and plus means to the left.

00:22 But let's just make sure we understand why.

00:25 So we know that this is going to be 5, this is going to be a shift 5 units to the right.

00:31 But why is that? imagine if we plugged zero into the original function, f of zero.

00:40 And that's just going to equal some random thing, whatever.

00:44 Let's just call it a square for now.

00:45 It could be any number.

00:46 Who knows? when i plug zero into this new function, i'm going to get f of 0 minus 5, which is going to be f of negative 5.

01:06 So zero has actually been moved five spaces to the right, because before when i would plug zero in, it was just whatever this number is.

01:21 But now when i try to plug zero in, it's taken me all the way to f of negative five.

01:28 It's almost like if here's my number line, here's zero, and here's negative five.

01:34 This used to be where zero is, but now negative 5 has shifted all the way over here, and it's in the new place where 0 used to be.

01:46 So then what do you think happened to 0? it shifted, 5 spaces over as well.

01:53 So now, to get this square that i used to get by plugging 0 in, now i would have to plug 5 in to get that square...

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