Which of the following describes the graph of y = |x| after it has been reflected across the
x-axis, shifted vertically up 3 units, and shifted horizontally to the left 6 units?
a) y = x + 6 + 3
b) y=x+61 +3
c) y = x 6 + 3
d) y = x 6 + 3
The correct answer is c) y = x - 6 + 3.
When a graph is reflected across the x-axis, the sign of the y-coordinate is changed. So, the equation becomes y = -|x|.
When the graph is shifted vertically up 3 units, this means adding 3 to the y-coordinate. So, the equation becomes y = -|x| + 3.
When the graph is shifted horizontally to the left 6 units, this means subtracting 6 from the x-coordinate. So, the equation becomes y = -|x-6| + 3.
Simplifying, we get y = x - 6 + 3, which is option c).
To determine the graph of y = |x| after it has been reflected across the x-axis, shifted vertically up 3 units, and shifted horizontally to the left 6 units, we can follow these steps:
1. Reflection across the x-axis: When a function is reflected across the x-axis, the sign of the y-values changes. So, the absolute value |x| becomes -|x|.
The equation becomes: y = -|x|
2. Vertical shift up 3 units: Shifting the function vertically by a positive number moves the entire graph up by that amount. So, to shift the graph up 3 units, we add 3 to the function.
The equation becomes: y = -|x| + 3
3. Horizontal shift to the left 6 units: Shifting the function horizontally to the left moves the graph to the left by the specified amount. To shift the graph left 6 units, we replace x with (x + 6) to indicate that x is increased by 6.
The equation becomes: y = -|x + 6| + 3
Therefore, the correct option is d) y = -|x + 6| + 3.
The correct answer is:
d) y = |x + 6| + 3