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

Key Concepts

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.

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.

Transcript

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

00:05 The first function, f of x, is 2 to the x.

00:10 And the second function, g of x, is 2 to the x minus 1.

00:16 Let's start by making a coordinate table.

00:18 It gives us the coordinates negative 2, negative 1, 0, 1, and 2.

00:24 I'll start with f first.

00:28 So when x is negative 2, f of x is going to be 2 to the negative 2.

00:32 We know with negative exponents it takes something in the numerator and brings it to the denominator, so we'll have 1 over 2 squared.

00:42 Like this, we can see that if we solve for 2 squared, it's just 1 over 4.

00:48 The same thing happens with x equal negative 1.

00:52 We'll have 2 to the negative 1, which just brings it into the denominator, 2 to the 1st, which is just 1 half.

01:01 Next, when x is 0, f of 0 is going to be 2 to the 0.

01:05 By now, your rules for exponents should be telling you that this is just going to be one.

01:11 Anything raised to the 0th is just 1.

01:14 Another rule for exponents is that anything raised to the first power is just itself.

01:18 So, an x is 1, f of x, which is 2 to the 1st, is just going to be 2.

01:22 And finally, when x is 2, f of x is 2 squared or 4.

01:29 So those are all the fs.

01:30 Let's take a look at the gs now.

01:33 We can see that when x is negative 2 g of x is going to be 2 to the negative 2 minus 1.

01:40 Well, we already solved for 2 to the negative 2.

01:43 It's 1 4th.

01:44 So 1 4th minus 1, which is just going to be equal to negative 3 4ths.

01:51 Next, when we have x equal to 1, we'll have 2 to the negative 1 1 1, which we know is just 1⁄2 minus 1.

02:00 That we know is going to be negative 1⁄2.

02:04 When x is zero we'll have two to the zero minus one two to the zero minus one we know is one minus one which is zero two to the first minus one will be two minus one which is one and two squared minus one it's just going to be four minus one which is equal to three so now that we have the coordinates for both functions let's go ahead and draw some axes and we'll plot this so i will draw some axes here this is our x -axis, and here is our y -axis.

02:38 And i'll choose the y -axis.

02:42 We'll start at 4 and go it all the way down to negative 1.

02:47 So it'll be negative 1...

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