Graph functions f and g in the same rectangular coordinate...

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x-1}$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x-1}$$

Key Concepts

Horizontal Translations

A horizontal translation shifts the graph of a function left or right without changing its shape. In the function g(x)=2^(x-1), the x-1 inside the exponent indicates that the original graph of f(x)=2^x is shifted to the right by 1 unit.

Graphing Functions in a Rectangular Coordinate System

Graphing functions in a rectangular coordinate system involves plotting computed values for selected x-values and marking corresponding y-values to visualize the behavior of the function. By choosing specific integer values, one can clearly compare and relate the transformed graphs of different functions.

Exponential Functions

Exponential functions are functions in which the independent variable appears as an exponent. They are characterized by a constant base raised to a variable power, leading to rapid growth or decay. Key features include a horizontal asymptote, typically the x-axis, and a consistent multiplicative rate of change.

Function Transformations

Function transformations involve modifying a basic function in various ways such as translations, reflections, stretches, and compressions. Understanding these transformations helps in quickly determining how the graph of a basic function will change when alterations are made to its formula.

Transcript

00:01 This question gives us two functions and asks us to make a coordinate table as well as a graph.

00:05 And doing that, we'll see the relationship between them.

00:08 The first function it gives is f of x equals 2 to the x.

00:12 The second is g of x equals 2 to the x minus 1.

00:19 It asks us to use the coordinates x equals negative 2, negative 1, 0, 1, and 2.

00:27 So i'll start with f of x.

00:29 We'll start on the left side here.

00:33 When we plug in negative 2 for x, we'll get that f of negative 2 is 2 to the negative 2.

00:42 We know when we have negative exponents, we just flip from the numerator to the denominator and make the exponent positive.

00:47 So it's going to be 1 over 2 squared, which is the same thing as 1 fourth.

00:54 Next, when x is negative 1, we'll have the same sort of thing, 2 to the negative 1.

01:00 Again, we flip it, 1 over 2 to the 1st, which is the 1 half.

01:04 When x is 0, we'll have 2 to the 0.

01:07 Now anything to the 0 by now we know is just 1.

01:11 Then when x is 1, we'll have 2 to the 1.

01:13 2 to the 1st, we know anything to the 1st is just itself, so 2 to the 1st is 2.

01:20 And then finally, we have 2 to the 2.

01:23 2 squared is just 4.

01:26 So these are 5 coordinates in f of x, now we'll look at g of x.

01:31 When x is negative 2, g of x is 2 to the negative 2 minus 1, which is the same as 2 to the negative 3.

01:40 Now, again, we'll move from the numerator to the denominator and change the side of the exponent.

01:45 We'll have 1 over 2 thirds, and 1 over 2 to the 3, and 2 to the 3 we know is 8, so we'll have 1 over 8.

01:54 So, g of negative 2 is 1 8.

01:59 When x is negative 1, we'll have 2 to the negative 1 minus 1, which is the same as 2 to the negative 2.

02:06 And that's something we already solved for up here in f of negative 2.

02:10 2 to the negative 2 we know is just 1 fourth.

02:14 Next, we have 2 to the 0 minus 1, which is the same as 2 to the negative 1.

02:20 Again, already something we found.

02:22 We just know that that's 1⁄2.

02:25 We'll keep moving.

02:26 We'll get 2 to the 1 minus 1, which is 2 to the 0.

02:31 Again, something we know, which is just 1.

02:35 And finally, g of 2 is 2 to the 2 minus 1, which is 2 to the 1.

02:40 We found this in f of 1.

02:42 It's just 2.

02:44 So now we have two sets of coordinates, and i'm going to scroll down here to draw some axes where we can plot them.

02:51 Draw some axes...

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