Answer:
Please see the attached graph. Graph y = -f(x - 4) - 1 is shown in red.
Explanation:
Graph transformations involve modifying the shape, position, or orientation of a graphed function by applying operations such as translations, reflections, stretches, or compressions.
The graph of y = -f(x - 4) - 1 is obtained by transforming the graph of y = f(x) through a series of translations and a reflection as follows:
- f(x) → f(x - 4): This is a translation of 4 units to the right.
- f(x - 4) → -f(x - 4): This results in a reflection of the graph in the x-axis.
- -f(x - 4) → -f(x - 4) - 1: This is a translation of 1 unit down.
The points of the graph of y = f(x) are:
- (-4, -2)
- (0, 2)
- (2, 2)
- (4, 0)
Translation of 4 units to the right f(x) → f(x - 4):
- (-4, -2) → (0, -2)
- (0, 2) → (4, 2)
- (2, 2) → (6, 2)
- (4, 0) → (8, 0)
Reflection in the x-axis f(x - 4) → -f(x - 4):
- (0, -2) → (0, 2)
- (4, 2) → (4, -2)
- (6, 2) → (6, -2)
- (8, 0) → (8, 0)
Translation of 1 unit down -f(x - 4) → -f(x - 4) - 1:
- (0, 2) → (0, 1)
- (4, -2) → (4, -3)
- (6, -2) → (6, -3)
- (8, 0) → (8, -1)
To graph y = -f(x - 4) - 1, plot the transformed points (0, 1), (4, -3), (6, -3) and (8, -1) and connect them with straight lines.