Problems 87-90 are based on material learned earlier in the course. The purpose of these problem

step 1 and it's going to be graphing a function. All right. So here's the function that we're given. F

Problems 87-90 are based on material learned earlier in the...

Key Concepts

Exponential Functions

Exponential functions are defined by expressions in which the variable appears in the exponent. They typically exhibit rapid growth or decay and have characteristic features such as a horizontal asymptote. Understanding these functions includes knowing their domain, range, and behavior as the input values become very large or very small.

Function Transformations

Function transformations involve shifting, reflecting, stretching, or compressing the graph of a base function. For exponential functions, these transformations are applied through modifications both outside and inside the exponent. External factors can vertically shift or reflect the graph, while changes inside the exponent generally result in horizontal shifts or reflections.

Graphical Analysis and Sketching

Graphical analysis involves identifying key features of the function such as intercepts, asymptotes, and important points resulting from transformations. Sketching the graph of a transformed exponential function requires first understanding the behavior of the base function and then systematically applying the transformations to locate critical points and determine the overall shape of the graph.

Transcript

00:01 And it's going to be graphing a function.

00:03 All right.

00:04 So here's the function that we're given.

00:09 F of x is equal to negative 3 to the power of 1 minus x plus 2.

00:15 So the first step we're going to want to do is find the domain and range, which i've done right here.

00:20 And when you plug in random numbers for the x and substitute it in, you will find out that this function will be true.

00:33 For all of the x values.

00:35 So it's going to go from negative infinity to positive infinity.

00:38 In the range, we have the y intercept right here.

00:42 So we know that it's not going to be passing two.

00:46 And that's going to be your limit right there.

00:48 But it will be going below as much as it could.

00:53 So that's a negative infinity to two.

00:55 And after that, we will be finding the x intercepts and the y intercepts, which i have done right here as well, to save us some time.

01:05 And this is where differentiation comes in.

01:08 So we're going to go ahead and plug in the natural log of 2, negative natural log of 2 plus negative natural log of 3, all over the natural log of 3.

01:18 And if you think about it, the 3 on the, the natural log of 3 on the top and the natural log of 3 on the bottom will be canceling out and leaving us with the negative natural log of 2, which is actually going to give us 0 .369, or approximately 0 .369.

01:43 So it's good to write that down.

01:51 And the x value is going to be 0, or the y value is going to be 0.

01:55 And in this situation, the y value is going to be 0.

01:57 And the y intercept is actually 1, negative 1, because we're going to be starting out at negative 3.

02:04 And then plus 2 will give us negative 1.

02:08 And after finding the domain range x intercept and y intercept, we're going to want to find the horizontal asymptote, vertical asymptote, and extreme points.

02:21 There are no extreme points and there are no vertical asymptotes, but there is a horizontal asymptote which is y equals 2.

02:28 All right, now let's get into graphing.

02:31 And what does the horizontal asymptote mean? it means that the graph will not go past my resume.

02:38 And since we'll be going to the negative side, let me go ahead and create my margin right about here...

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