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

Name: Problem 37, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2021-06-15T23:14:13-08:00
Duration: 4 min 28 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)

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

Key Concepts

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.

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.

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.

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.

Transcript

00:01 Number 37 in section 8 .1 is giving us a graph of an exponential curve.

00:08 This one is flat on the left end, steep on the right end.

00:13 So this is an exponential growth curve.

00:19 And i'll remind you that when we're looking at growth, the base for your exponent, they're calling it the a value in their form.

00:29 I call it the b value.

00:30 That has to be a number that's big.

00:33 Bigger than one.

00:36 All right.

00:36 So this base for our exponent we want it to be bigger than one.

00:42 And we're going to write our equation because we've shifted this function up from the x -axis and an apparent graph for an exponential function.

00:54 Hopefully you remember that the graph doesn't go into the negative y values.

01:00 So we've shifted this graph up and we've got an ascentral function.

01:05 Horizontal asymptote at the value y equals one, which is actually in this form that we're going to write our equation, that's that c value.

01:15 Remember, that's your vertical shift in this equation.

01:20 So the first thing that we can do is adjust our function and make sure that that c value is one.

01:30 And then since we know three points that are on this graph, we're going to take each point or a couple of these points and we're going to substitute them in.

01:43 And i would probably start with this point right here because working with a zero is always easiest.

01:52 So remember, f of x is another way to represent the y value.

01:55 So we put three and four the y...

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