Find an exponential function of the form f(x)=b a^x+c for the graph. (graph can't copy)

Name: Problem 38, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2021-06-15T23:19:53-08:00
Duration: 4 min 46 s
Channel: Michelle Alfonso
Description: Find an exponential function of the form f(x)=b a^x+c for the graph. (graph can't copy)

step 1 Hi, for number 38, we're working with an exponential function here. And they give us three known

Find an exponential function of the form f(x)=b a^x+c for...

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent, typically written in the form f(x) = b * a^x or f(x) = b * a^x + c. These functions are widely used to model processes that involve growth or decay, and understanding their structure is essential for interpreting their behavior and graph characteristics.

Parameters in Exponential Models

In an exponential function, the parameters 'a', 'b', and 'c' play vital roles. The base 'a' determines the rate of growth (if a > 1) or decay (if 0 < a < 1). The coefficient 'b' affects the initial value and the vertical stretch or compression of the graph. The constant 'c', when present, translates the graph vertically, thus altering the horizontal asymptote.

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the basic exponential function. Vertical shifts, determined by the constant 'c', can raise or lower the graph, while horizontal or vertical stretches and compressions modify the spread and steepness. Recognizing these transformations is crucial when matching a function to a particular graphical representation.

Horizontal Asymptote

Exponential functions, especially those of the form f(x)=b*a^x+c, often feature a horizontal asymptote. This asymptote is the line that the graph approaches but never reaches, and it is defined by the value of 'c' in the function. Understanding asymptotic behavior is important for accurately sketching and analyzing exponential graphs.

Transcript

00:01 Hi, for number 38, we're working with an exponential function here.

00:08 And they give us three known points on this exponential function.

00:12 You can see that it's decreasing to the right.

00:16 So this would be an exponential decay function.

00:20 And they're asking us to take the information from this graph that we know and write an equation for it in this form.

00:28 F of x equals b times a to the x.

00:32 X power plus c so since we know that this is a decay function we know the base for that exponent the a value is going to have to be a number that's in between zero and one that's how we can tell that a function is representing exponential decay the other thing we know is this c value the number that's added after our exponential part that represents where the horizontal asymptote is.

01:05 And if you remember your parent function for an exponential function, that horizontal asymptote usually is right along the x -axis.

01:14 We can see that it's been pushed up.

01:17 And i can tell that, you know, my y values are getting smaller and smaller, but they're probably never going to be smaller than y equals one.

01:30 So this is going to be the c value in our equivalent.

01:33 So our first step, i'm going to change my f of x to the simpler form, just a y.

01:41 We're going to plug in that c value.

01:44 Now we're still looking for these two parameters to complete our equation, the coefficient b, and the base for our exponent, the a value.

01:55 And there's a reason they give us three other points here.

01:59 We can start substituting some things in.

02:02 So when i work with these kinds of functions, i always like to start with an easy point.

02:11 And the easiest point i see is 03.

02:13 So i'm going to start there and i'm going to plug in the point 0 .03 for x and y.

02:21 And let's see what happens.

02:25 Oh, i like this because when i have a base raised to the zero power, remember any base to the zero power just makes that.

02:35 Value 1...

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