The graph of f is given. Draw the graphs of the following functions. (a) y=f(x)-2 (b)

step 1 In this problem, we are given the graph of a function f. We are asked to draw the graphs of func

The graph of f is given. Draw the graphs of the following...

Key Concepts

Function Transformations

Function transformations involve modifying the graph of a given function in various ways such as translating (shifting) it vertically or horizontally, scaling it vertically or horizontally, and reflecting it about an axis. These transformations allow one to predict how the graph of a function will change when constants are added, subtracted, multiplied, or applied to the input variable.

Vertical Translation

Vertical translation shifts the entire graph of a function up or down without changing its shape. This is achieved by adding or subtracting a constant to the function's output, as in y = f(x) + k, where a positive k moves the graph upward and a negative k moves it downward.

Horizontal Translation

Horizontal translation shifts the graph of a function left or right without altering its overall shape. This translation is performed by adding or subtracting a constant to the variable inside the function, as in y = f(x - h), where a positive h shifts the graph to the right, and a negative h shifts it to the left.

Vertical Scaling and Reflection

Vertical scaling involves multiplying the output of a function by a constant, which stretches or compresses the graph in the vertical direction. If the constant factor is greater than 1, the graph is stretched; if it is between 0 and 1, it is compressed. When this constant is negative, it also reflects the graph across the x-axis, thereby inverting the peaks and valleys of the function.

Horizontal Scaling

Horizontal scaling is performed by multiplying the input variable by a constant, which compresses or stretches the graph along the horizontal axis. In the transformation y = f(ax), if |a| > 1 the graph is compressed horizontally, while if 0 < |a| < 1 the graph is stretched horizontally. This transformation changes the speed at which the function's outputs vary with respect to the input.

Transcript

00:02 In this problem, we are given the graph of a function f.

00:07 We are asked to draw the graphs of function which are obtained from f.

00:15 In part a, we're given that function y is equal to f of x minus 2.

00:21 I'll be drawing this function in red.

00:26 So we see that this is going to be a shift of the graph of f down by two units.

00:44 So in particular, you know that a negative thing, or i guess just negative 2, to 1 on the x -axis, we're shifting it's down 2 units.

00:56 So this is going to move down 2 over here, and this will move down to 0.

01:05 So you have a line that's parallel, and then between 1 and 2, we're going to have the value of 0 for the function, and then between two and three, we're going to have another downward slope and line just parallel to the one above it.

01:32 And this is, we'll call this y1.

01:40 So this is function y1 in part b.

01:51 We're given that y is the function f of x minus 2.

01:58 You see that this is a shift of the graph of f by two units and because it's negative 2 to the right.

02:22 I'm going to draw this graph and you bring up.

02:26 I'm going to call it y2 and we'll see that going back to original graph f that means from x equals negative 2 to x equals 1 or to shift this line over to another parallel line.

03:02 So in particular, at x equal 1, we have then f of x is 2, so y2 of 3 is going to be 2, down one slope and line.

03:21 And the domain of this function is going to be starting at 0 instead of at negative 2.

03:30 That's part of it.

03:32 And then we see from x equals 3, we're going to have the same as the graph of f from one on.

03:50 So particular, we're going to move over one unit, and then we'll move over, down, one unit.

04:05 And this is going to be y2.

04:12 In part c, we're given that y is the function negative 2, f of a very 1 .5 .2.

04:21 This is a two different transformations.

04:27 So we have first stretch a graph vertically by a factor of two, then we reflect graph about the x -axis because of the negative signs.

05:09 And we'll call this one y3, draw it in blue.

05:13 So first things first, you want to take the function of that and stretch it.

05:26 So in particular, picking a few points here, we have at x equals negative 2, f of x is negative 1.

05:38 And so we're going to stretch that down to negative 2.

05:43 Then flip it across the axis so that the point becomes two, the value is two.

05:52 And because of reflection, we are drawing here the value that is two, so we double that, get four, and then flip it.

06:33 And we'll connect these two points by line.

06:38 And then from x equals 1, x equals 2, we have the f is 2.

06:45 So we double that, we get 4 again.

06:47 And then you flip it, we get another straight line there...

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