Graph functions f and g in the same rectangular coordinate system. Select integers from -2 to 2

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

Key Concepts

Vertical Scaling

Vertical scaling entails multiplying the entire function by a constant. This operation stretches the graph vertically if the constant is greater than one or compresses it if the constant is between zero and one, thus altering the steepness of the function.

Vertical Shifts

Vertical shifts are implemented by adding to or subtracting from the entire function. Subtracting a constant results in shifting the graph downward, adjusting the output values uniformly without changing the overall shape.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, leading to rapid growth or decay. They typically have a horizontal asymptote and demonstrate consistent multiplicative change. Understanding their standard behavior is fundamental when applying transformations.

Function Transformations

Function transformations involve systematically altering a function's graph through operations such as translation, stretching, and reflection. These operations help relate the graph of one function to another by shifting its position or changing its shape.

Horizontal Shifts

A horizontal shift occurs when a constant is added to or subtracted from the input variable within the function. Adding a positive constant inside the function (as in f(x + 1)) moves the graph to the left by that constant value, altering the input’s effect on the output.

Transcript

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

00:06 The functions that it gives are f of x equals 2 to the x and g of x equals 2 to the x plus 1 minus 1.

00:20 Now it gives you some x to use.

00:23 It wants you to go from negative 2, negative 1, 0, 1, all up to 2.

00:29 So let's start.

00:30 I'm going to start with the f's first, and then we'll go to the g's.

00:34 When x is negative 2, f of x is 2 to the negative 2.

00:38 We know with negative exponents, you switch from the numerator to the denominator and change the sign on the exponent.

00:44 So it's going to look like 1 over 2 squared, which is 1 fourth.

00:50 Similarly, when x is negative 1, we'll have 2 to the negative 1, which is just 1 over 2 to the 1⁄2.

00:55 Which is just one half.

00:58 And when x is zero, we'll have two to the zero.

01:00 By now we know that anything raised to the zero is just one.

01:04 When x is one, we'll have two to the first.

01:06 Now, anything raised to the first is just itself, so it's just going to be two.

01:11 And when x is two, we'll have two raised to the second, which is equal to four.

01:17 So these are five coordinates on the f graph, and now let's look at the g side.

01:22 When x is negative two, we'll have two to the negative two plus one, minus one.

01:28 Now 2 to the negative 2 plus 1 is the same as 2 to the negative 1 plus 1.

01:36 Now we know that 2 to the negative 1 first is just 1 half, so this is going to be, oh excuse me, this is minus 1, isn't it? so this is minus 1.

01:45 So we'll be left with 1⁄2 minus 1, which is just negative 1 half.

01:52 Next we'll have 2 to the negative 1 plus 1 minus 1, which is 2 to the 0 minus 1.

02:00 We know 2 to the 0 is 1, so it's going to be 1 minus 1, which is just 0.

02:07 Then we'll have 2 to the 0 plus 1, which is minus 1, which is just 2 to the 1 1 1 1 1 which we know is 2 minus 1, which we know is 2 minus 1, which will be 1.

02:20 Then we will have 2 to the 1 plus 1 minus 1.

02:25 In other words, 2 is squared minus 1.

02:28 Now 2 squared we know is 4, so 4 minus 1.

02:33 And g of 1 will be 3.

02:36 And finally, when x is equal to 2, we'll have 2 to the 2 plus 1 minus 1, which is 2 cubed minus 1.

02:44 We know the 2 cubed is 8.

02:46 So 8 minus 1, g of 2 will be 7.

02:50 So now that we have all of this, let's go scroll down and try to plot some of it...