We should check for each type of transformation: translations (vertical and/or horizontal), dilation (either compression or stretch), and reflection.

We can see that there is no reflection across the x-axis since the sign outside the absolute value symbol is positive.

There is also no dilation because there is no coefficient outside of the absolute value symbol.

Inside the absolute value sign we have x-1. The -1 represents a horizontal translation of 1 unit to the right.

Outside the absolute value symbol we have +2. This represents a vertical translation of 2 units upward.

We can start with the graph of y = \left|x\right| and use the transformations found in the previous part to graph the function f\left(x\right) = \left|x - 1\right| + 2.

We know that if we start from y = \left|x\right|, we need to move 1 unit to the right, and 2 units upward. This results in the following graph:

The vertex of an absolute value function is its maximum or minimum point. We can see that this function has a minimum at \left(1, 2\right) by looking at the graph.

The vertex of an absolute value function in the form f\left(x\right)=\pm a\left\vert x-h\right\vert +k is always \left(h,k\right).

Remember that the domain is the set of all x-values that can be put into the function, while the range is the set of all function values (i.e. y-values) that can be obtained by the function.

Looking at the graph, we can see that the domain will be "all real values", as any value of x can be used as an input.

The vertex of this function is at \left(1, 2\right) and is a minimum point, so the range will be "all values greater than or equal to 2."

Using interval notation, we have:

• Domain: \left(-\infty, \infty\right)
• Range: \left[2, \infty\right)

We should check for each type of transformation: Translations (vertical and/or horizontal), dilation (either compression or stretch), and reflection.

It may help to add the graph of y = \left|x\right| to the same coordinate plane:

We can see that the graph has a maximum point rather than a minimum point, so there has been a reflection across the x-axis.

The vertex of the graph is at \left(-3, 0\right). This is 3 units to the left of the vertex of y = \left|x\right| and at the same y-value. So there has been a horizontal translation of 3 units to the left, and no vertical translation.

We can also see that the rate of change of the two graphs are different. The slopes of the sides of this function are 2 and -2, while the slopes of y = \left|x\right| are 1 and -1. So there has been a vertical stretch by a factor of 2.

For linear and absolute value functions, a changed slope indicates a dilation. In this case, we stated that it was a vertical stretch with a factor of 2, but it could equivalently be thought of as a horizontal compression with a factor of \dfrac{1}{2} (as long as the compression happens before the horizontal translation).

We can use the transformations that we described in part (a) and apply them to the function y = \left|x\right| to get an equation for the function shown.

Starting with y = \left|x\right|, we can apply the transformations one at a time:

So the equation of the function is y = -2\left|x + 3\right|.

It is important to note that the order of transformations matters in some cases. For example, applying a vertical translation and then a reflection across the x-axis will be different to reflecting first and then applying the same vertical translation.

In this case, the transformations of a vertical stretch, a reflection across the x-axis, and a horizontal translation are all independant and can be applied in any order.