How do you graph f(x) = -2^(x+3) + 4. Can you explain?

Question

I know x+3 is a shift to the left, so we subtract 3 from each original x point (what are the original x and y points?).

I know +4 is the constant, so it’s the horizontal asymptote? How do I know it’s not the vertical asymptote??

I know the -2 means I multiply -2 by each original y point.

Am am I correct so far? Please help me finish and understand. Thanks.

Answer 1

The first question to ask is "What is the parent function?" because that will give you the "original" x and y values. Here, the parent function is y = 2^x which looks like this:

Let's start with the point (0, 1), move it left by 3 to make it (-3, 1). The "-" in front of the 2 will reflect the graph over the x axis so now the point becomes (-3, -1). Now, move the entire graph up by 4 making the point (-3, 3).

You can do that same for all the points (or enough to get the idea of how the graph goes).

Answer 2

adding the 4 to the functions makes the graph shift UP 4 units

Answer 3

Your last statement is not right. The equation as written says: "Take the value of X and add 3. Then use the value as the exponent applied to 2. Multiply the result by -1, and then add 4"

To graph this, first let X = 0. Then Y = - 2³ + 4 = -8 + 4 = -4

For large positive values of X, Y heads down towards - ∞

For large negative values of X, the first expression becomes - ( 1 /2^(X+4) ). At very large values of X, this approaches 0, so Y approaches +4

Use these points to sketch the plot, then evaluate for specific values to refine the curve.

You can check you work by using an online graphing tool---eg this:

https://www.desmos.com/calculator