Question

Describe the transformations from the graph of f(x)=∣x∣ to the graph of c(x)=∣x+4∣+4.

Answer 1

1) horizontal shift to the left by 4 units

2) vertical shift upwards by 4 units

Step-by-step explanation:

the transformations are two translations. the first affects the x, where x becomes x+4. this means a leftward shift parallel to the x-axis. the other affects the y, essentially shifting the graph up by 4 units for every y.

thus, we have the two translations above.

Answer 2

Transforming the graph of f(x)=|x| to c(x)=|x+4|+4 involves a leftward shift by 4 units and an upward shift by 4 units, resulting in the vertex of the V-shaped graph moving from (0,0) to (-4,4).

Step-by-step explanation:

The transformation from the graph of f(x)=|x| to the graph of c(x)=|x+4|+4 involves two main steps. The absolute function f(x) is known for its V-shaped graph with the point of the V at the origin (0,0). When we add 4 inside the absolute value, as in |x+4|, we are shifting this graph 4 units to the left. This is because we need to input a value for x that is 4 less than before to get the output to be zero—the bottom of the V moves from (0,0) to (-4,0). The second transformation involves adding 4 to the entire function, which results in shifting the graph 4 units upwards.

So, the sequence of transformations is a horizontal shift to the left by 4 units followed by a vertical shift upwards by 4 units. To help visualize this, consider the V-shaped graph of f(x)=|x|, which touches the x-axis at the origin. After the transformations, the graph of c(x) will touch the x-axis at (-4,0) instead and will be hovering above the x-axis, with its vertex at the point (-4,4).