Suppose (2,5) is a point on the graph of y=f(x). Fill in the blanks with the transformed coordin

Suppose (2,5) is a point on the graph of y=f(x). Fill in the...

Key Concepts

Composite Transformations

Composite transformations combine two or more basic transformations, such as translations and scalings. When applying composite transformations, it is important to account for the order of the operations, as the effect on the graph can vary depending on the sequence of applied transformations.

Horizontal Scaling

Horizontal scaling involves stretching or compressing the graph of a function along the horizontal axis. This is achieved by multiplying the input variable by a constant inside the function. The reciprocal of the constant determines the degree of stretch or compression, and it affects the x-coordinates of the points on the graph.

Vertical Scaling

Vertical scaling is the process of stretching or compressing the graph of a function in the vertical direction. This is accomplished by multiplying the function's output by a constant. If the constant is greater than one, the graph stretches; if it is between zero and one, the graph compresses.

Function Transformations

This concept involves applying changes to the graph of a function to produce a new graph. The changes can include shifting, stretching, compressing, or reflecting the graph. Understanding function transformations is fundamental for analyzing how modifications to function expressions affect their graphs.

Vertical Translation

Vertical translation refers to shifting the graph of a function up or down by adding or subtracting a constant from the function output. This transformation affects the y-coordinates of points on the graph without changing the x-coordinates.

Horizontal Translation

Horizontal translation involves shifting the graph left or right by adding or subtracting a constant inside the function’s argument. Unlike vertical translations, horizontal translations directly affect the x-coordinates of points on the graph.

Transcript

00:01 Alright, we're going to be tracking a point through some transformations to get a handle on transformations.

00:08 So suppose that 2 comma 5 is a point on the graph of an arbitrary function, call it f of x.

00:23 What does that mean? that means that 2 .5 has the form x comma f of x.

00:28 Or in other words, f of 2 is equal to 5.

00:36 So, if 2, 5 is on the graph of f of x, then what point can we say, for sure, is on the graph of f of x minus 4? supposing we knew nothing else about f of x.

00:50 We just know 2 .5 something, right? so 2 .5 is up here.

00:57 And we're going to draw the graph, what we know about the graph of f of x minus 4, so we're going to put an empty hole right here because we know that 2 .5 is from here, it must have ended up somewhere else.

01:10 Well, whenever you subtract 4, what you do is you are moving down vertically, right? you're moving down four units.

01:23 So the point that is on the graph of f of x minus 4 is actually 2 .1.

01:28 2 .5 became 2 .1.

01:35 And you'll notice that since the sort of subtraction was on the outside of the f of x, we ended up subtracting it from the output, right? so thinking about it geometrically is one way, but later on, we're going to run into more difficult situations where you probably want to think about it more symbolically.

01:56 So you should get a handle on both points of view.

02:00 So that was that.

02:02 What about a horizontal transformation, like f of x minus.

02:05 4 on the inside.

02:09 So don't forget, whatever you're dealing with vertical transformations, it works the way you would expect.

02:17 Minus means you move in the negative direction, positive means moves and move in the positive direction.

02:24 For horizontal transformations, whenever you are messing with x itself on the inside, it works the opposite.

02:32 So x minus 4 moves the 2 .5 point to the right in the positive direction to 6 .5, right? so 2 .5 becomes 6 .5.

02:51 We can say that 6 .5 is on the graph of f of x minus 4 quantity.

03:02 Right? so what about the transformation negative 7 f of x.

03:11 So we know geometrically that since we're taking the negative, it's going to reflect it around the x -axis, right? and then span it by 7.

03:23 But you might think about this more symbolically.

03:27 You could say that what this does is negative...

03:32 Since negative 7f of 2 is equal to negative 7 times 5, right? because that's what f of 2 is, that you get negative 35.

03:51 Right? so on the graph of this function, negative 7f, you know that you get like one little point right here, 2 .5.

04:03 That is not on the graph.

04:04 It has been moved, it has been reflected, and multiplied to become 2 comma negative 35...

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