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 option is d) 𝑦 = |𝑥 − 6| + 3.
When we reflect the graph of 𝑦 = |𝑥| across the x-axis, it becomes 𝑦 = −|𝑥|.
When we shift it vertically up 3 units, it becomes 𝑦 = −|𝑥| + 3.
When we shift it horizontally to the left 6 units, it becomes 𝑦 = −|𝑥 - 6| + 3.
Adding the absolute value function back to the equation gives 𝑦 = |𝑥 - 6| + 3.
The series of transformations applied to the graph of 𝑦 = |𝑥| are as follows:
1. Reflection across the x-axis: This transformation flips the graph upside down. The positive and negative y-values are reversed while the x-values remain the same.
2. Shift vertically up 3 units: This transformation moves the entire graph upward by 3 units. Every y-value in the original graph is increased by 3.
3. Shift horizontally to the left 6 units: This transformation moves the entire graph horizontally to the left by 6 units. Every x-value in the original graph is decreased by 6.
Therefore, the correct expression that describes the graph after all the transformations is 𝑦 =− |𝑥 − 6| + 3, which corresponds to option (c).
In summary, the correct answer is c) 𝑦 =− |𝑥 − 6| + 3.
To determine the correct answer choice, let's break down each transformation step and analyze how it affects the original function.
1. Reflection across the x-axis:
When a function is reflected across the x-axis, the sign of the function changes. For the function 𝑦 = |𝑥|, after reflection, it becomes 𝑦 = −|𝑥|.
2. Vertical shift up 3 units:
After the reflection, the function is shifted vertically up by 3 units. This means we need to add 3 to the function. Therefore, the transformed function becomes 𝑦 = −|𝑥| + 3.
3. Horizontal shift to the left 6 units:
To shift the function horizontally to the left, we need to subtract the shift amount from the 𝑥 variable. After the horizontal shift of 6 units to the left, the function becomes 𝑦 = −|𝑥 − 6| + 3.
Comparing the transformed function with the answer choices:
a) 𝑦 = | − 𝑥 + 6| + 3 - This choice does not reflect the original function across the x-axis.
b) 𝑦 =− |𝑥 + 6| + 3 - This choice reflects the function, but does not shift it horizontally to the left.
c) 𝑦 =− |𝑥 − 6| + 3 - This choice reflects the function and shifts it horizontally to the left.
d) 𝑦 = |𝑥 − 6| + 3 - This choice does not reflect the original function across the x-axis.
Therefore, the correct answer is c) 𝑦 =− |𝑥 − 6| + 3.