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

Key Concepts

Exponential Function

An exponential function is a mathematical function in which the variable appears in the exponent, typically expressed as f(x) = a^x where a is a positive constant not equal to 1. This type of function models rapid growth or decay and has important properties such as a constant relative rate of change and a horizontal asymptote.

Graphing Functions

Graphing functions involves plotting points corresponding to various values of x and connecting them to visualize the behavior of the function on the coordinate plane. It requires understanding key features such as intercepts, asymptotes, and intervals of increase or decrease.

Function Transformation

Function transformation refers to the process of modifying a function's graph through operations such as translations, reflections, stretches, and compressions. In this context, adding a constant to the input of the function causes a horizontal shift, while adding a constant to the function's output causes a vertical shift, effectively moving the graph without altering its overall shape.

Transcript

00:01 This question gives you two functions and asks you to create a coordinate table as well as a couple graphs.

00:06 The function it gives, the functions it gives are this.

00:11 F of x equals 2 to the x, and g of x equals 2 to the x plus 1 minus 2.

00:21 It also asks you to use the coordinates x equals negative 2 all the way up to positive 2, inclusive.

00:29 So i'm going to start with x equals negative 2, and i'll do all the fs.

00:35 So first, when x equals negative 2, we have f of x equals 2 to the negative 2.

00:39 We know with negative exponents, we just move the exponential term from the numerator to the denominator and take the negative off, so it'll be 1 over 2 squared.

00:50 This we can simplify to 1 fourth.

00:53 Next, when x equals negative 1, we'll have 2 to the negative 1.

00:56 Again, we just move the exponential term to the denominator.

01:00 We'll have 1 over 2 to the 1st, which is the same thing as 1 half.

01:04 Then when x is 0, we'll have 2 to the 0, and by now we know anything, except for 0, ris to the 0 of power is just 1.

01:13 Moving on, we have x equals 1.

01:15 F of x will be 2 to the 1st.

01:18 Now anything to the first is just itself, so 2 to the 1st will just be 2.

01:22 And finally, we have when x equals 2, f of x is 2 squared, and we know that 2 squared is 4.

01:29 So these are our 5 f coordinates.

01:32 Now let's take a look at g.

01:33 When x equals negative 2, g to the x equals 2 to the negative 2 plus 1 minus 2.