The graph of g(x) is a transformation of the graph of f(x)=3^x.
Enter the equation for g(x) in the box.
|Plots on graph (0,1) and (-1,-1)|
g(x) =
how about 3^(x+1) - 2 (shift left and down)
or 3*3^x - 2 (dilate and shift down)
g(x) = 3^(x+1) - 1
To determine the equation for g(x) based on the given information, we need to identify the type of transformation applied to the original function f(x)=3^x.
From the given plot points, we can observe that the graph of g(x) passes through (0,1) and (-1,-1). This indicates that there has been a shift in the y-direction.
Considering that the y-coordinate of both points has decreased by 2 units, it implies a vertical translation downward.
Therefore, the equation for g(x) can be written as:
g(x) = 3^x - 2
To find the equation for g(x), which is a transformation of the graph of f(x) = 3^x, we need to identify the specific transformation applied to f(x).
From the information given, we know that the graph of g(x) passes through the points (0,1) and (-1,-1). Let's use these points to determine the transformation of f(x).
First, let's consider the point (0,1). In the equation f(x) = 3^x, when x = 0, f(x) equals 1. So, it seems like the graph of g(x) has been shifted upward by 1 unit.
Next, let's consider the point (-1,-1). In the equation f(x) = 3^x, when x = -1, f(x) equals 1/3 (-1 as an exponent flips it to its reciprocal). So, it seems like the graph of g(x) has been reflected over the x-axis.
Based on this analysis, we can conclude that the transformation of f(x) to g(x) consists of an upward shift of 1 unit and a reflection over the x-axis.
Now, let's write the equation for g(x) using this information.
Start with the equation for f(x): f(x) = 3^x.
To reflect f(x) over the x-axis, we multiply f(x) by -1: -f(x) = -3^x.
To shift the graph upward by 1 unit, we add 1 to -f(x): g(x) = -3^x + 1.
So, the equation for g(x) is g(x) = -3^x + 1.