Fill in the blank with the appropriate axis (x -axis or y -axis). (a) The graph of y=-f(x) is ob

Fill in the blank with the appropriate axis (x -axis or y...

Key Concepts

Function Transformations

Function transformations involve systematically modifying a function's formula to analyze how its graph shifts, stretches, reflects, or compresses. This concept includes operations such as translations (shifts), reflections, and dilations that help in understanding the geometric impact on the graph when the function's equation is altered.

Reflection Transformation

Reflection transformation is a specific type of function transformation where the graph of a function is flipped over a specified line, typically one of the coordinate axes. Multiplying the output of a function by -1 results in a reflection over the x-axis, while replacing the input with its negative leads to a reflection over the y-axis. These reflections change the orientation of the graph in the coordinate plane without altering its shape.

Transcript

00:01 All right, so to answer this question, we need to figure out what is happening when we take some function f of x and then we just put a negative in front of it.

00:13 So if y is equal to negative f of x, let's see what happens by looking at this first table here.

00:20 So i've put my x coordinates around the origins.

00:24 We've got negative 2, negative 1 ,0, 1, 2, and i've given us some letters as our outputs.

00:30 And this is to show you that it really doesn't matter what the actual numbers are.

00:35 I just want you to focus on what happens to the numbers.

00:39 So the input negative 2 gave me the output a in the original function.

00:48 In the new function, where all i've done is multiply the output by a negative, all that's going to happen is this a is going to become a negative a.

01:02 Because all negative f of x means is you take that output f of x whatever it is in our case it was a and then you multiply it by a negative sign so now it's just negative a that's all that happens same thing with b it was the output was b but then it becomes negative b when we multiply f of x by that negative sign and again that happens with c that happens with d and that happens with e.

01:35 So the only thing negative f of x does, the only thing that negative sign does is it takes our outputs, our y coordinates, and it makes them negative.

01:47 So if it's affecting our y coordinates in this way, then the axis that it's going to reflect over is going to be the y -axis.

01:56 So that's why this is a reflection over the y -axis.

02:06 Axis.

02:09 Now when we look at this next situation, we have, this is just y equals f of negative x.

02:17 So let's see what's happening here.

02:20 This used to be, we have the same coordinates, negative 2, negative 1, 0, 1, and 2, which give us the outputs a, b, c, d, and e.

02:28 So if in this next column i have f of negative x, well, that means that i used to, in this very first column, sorry, in this first row here, what i was doing before over here was i plugged two in for x, and that is what gave me a.

02:52 But now, since i'm plugging in negative x, i'm not going to be plugging two in anymore.

02:58 Or sorry, i'm not going to be plugging in negative two in anymore.

03:02 I'm actually going to be plugging in two.

03:05 Because if i used to, i used to, i'm going to be plugging in two in anymore.

03:08 I'm actually going to be plugging in two...

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