Starting with the graph of
f(x)=3^x , write the equation of the graph that results from
(a) shifting
f(x) 1 units downward.
y=
(b) shifting
f(x) 1 units to the right.
y=
(c) reflecting
f(x) about the y-axis.
y=
(a) Shifting f(x) 1 unit downward:
Hmm, it seems like f(x) wants to go on a downward journey. But don't worry, we can help it with a little push! To make f(x) go 1 unit downward, we need to subtract 1 from the original equation.
So, the equation of the graph that results from shifting f(x) 1 unit downward is:
y = 3^x - 1
(b) Shifting f(x) 1 unit to the right:
Oh, f(x) wants to go for a little walk to the right! To give it a nudge in that direction, we need to replace x in the original equation with (x - 1). This will move f(x) 1 unit to the right.
Therefore, the equation of the graph that results from shifting f(x) 1 unit to the right is:
y = 3^(x - 1)
(c) Reflecting f(x) about the y-axis:
Now, f(x) wants to see its reflection in the mirror! It's time for a little y-axis transformation. To achieve this, we need to replace x with (-x) in the original equation.
As a result, the equation of the graph that results from reflecting f(x) about the y-axis is:
y = 3^(-x)
(a) Shifting f(x) 1 unit downward:
To shift the graph of f(x) = 3^x 1 unit downward, we need to subtract 1 from the function. Therefore, the equation of the resulting graph is:
y = 3^x - 1
(b) Shifting f(x) 1 unit to the right:
To shift the graph of f(x) = 3^x 1 unit to the right, we need to replace x with (x - 1) in the function. Therefore, the equation of the resulting graph is:
y = 3^(x - 1)
(c) Reflecting f(x) about the y-axis:
To reflect the graph of f(x) = 3^x about the y-axis, we need to replace x with (-x) in the function. Therefore, the equation of the resulting graph is:
y = 3^(-x)
To find the equations of the graphs resulting from shifting and reflecting the function f(x) = 3^x, we need to understand how these operations affect the original function.
(a) Shifting f(x) 1 unit downward:
To shift a function downward, you subtract a constant from the original function. In this case, we are shifting f(x) 1 unit downward. This means that every y-coordinate of the function will be 1 unit lower than the corresponding y-coordinate of f(x).
So, to find the equation of the shifted graph, we subtract 1 from the original function f(x) = 3^x:
y = 3^x - 1
This equation represents the graph resulting from shifting f(x) 1 unit downward.
(b) Shifting f(x) 1 unit to the right:
To shift a function to the right, you subtract the shifting value from the input variable. In this case, we are shifting f(x) 1 unit to the right. This means that every x-coordinate of the function will be 1 unit higher than the corresponding x-coordinate of f(x).
So, to find the equation of the shifted graph, we replace x with (x - 1) in the original function f(x) = 3^x:
y = 3^(x - 1)
This equation represents the graph resulting from shifting f(x) 1 unit to the right.
(c) Reflecting f(x) about the y-axis:
To reflect a function about the y-axis, you change the sign of the input variable. In this case, we are reflecting f(x) about the y-axis. This means that every x-coordinate of the function will have its sign reversed.
So, to find the equation of the reflected graph, we replace x with (-x) in the original function f(x) = 3^x:
y = 3^(-x)
This equation represents the graph resulting from reflecting f(x) about the y-axis.
(a) 3^x - 1
(b) 3^(x-1) - 1
(c) 3^(-(x-1))