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 + 6| + 3
c) 𝑦 =− |𝑥 − 6| + 3
d) 𝑦 = |𝑥 − 6| + 3
The correct answer is d) 𝑦 = |𝑥 − 6| + 3.
To describe the transformations applied to the function 𝑦 = |𝑥|, we need to consider the reflection across the x-axis, the vertical shift, and the horizontal shift.
1. Reflection across the x-axis: This can be achieved by multiplying the function by -1. The absolute value function is symmetric about the x-axis, so this transformation will reflect it across the x-axis.
2. Vertical shift up 3 units: We add 3 to the function to shift it vertically upwards.
3. Horizontal shift to the left 6 units: We subtract 6 from the x-value to shift the function horizontally to the left.
Given these transformations, the correct equation for the graph after the described transformations is:
b) 𝑦 =− |𝑥 + 6| + 3
To determine the answer, let's break down the problem step by step:
1. First, let's start with the graph of 𝑦 = |𝑥|. This is a simple V-shaped graph that is symmetric with respect to the y-axis.
2. The next step is to reflect the graph across the x-axis. When we reflect a graph across the x-axis, all the y-coordinates become their opposite, creating a downward reflection. So, the graph becomes 𝑦 = -|𝑥|.
3. Next, we need to shift the graph vertically up 3 units. When we shift a graph vertically, we add or subtract a value to the y-coordinate. In this case, since we want to shift it up, we add 3 to the graph. So, the graph becomes 𝑦 = -|𝑥| + 3.
4. Finally, we need to shift the graph horizontally to the left 6 units. When we shift a graph horizontally, we add or subtract a value to the x-coordinate. In this case, since we want to shift it to the left, we subtract 6 from the graph. So, the final transformed graph becomes 𝑦 = -|𝑥 - 6| + 3.
Therefore, the correct answer is (c) 𝑦 = − |𝑥 − 6| + 3.