Question

without graphing, identify the vertex, axis of symmetry, and transformations from the parent function f(x) = |x|. y = |x - 4| + 5. the vertex is (type an ordered pair.)

Explanation:

Step1: Recall vertex - form of absolute - value function

The general form of an absolute - value function is $y=a|x - h|+k$, and its vertex is $(h,k)$. For the function $y = |x - 4|+5$, we have $h = 4$ and $k = 5$.

Step2: Determine the vertex

The vertex of the function $y=|x - 4|+5$ is $(4,5)$.

Step3: Find the axis of symmetry

The axis of symmetry of the absolute - value function $y=a|x - h|+k$ is the vertical line $x = h$. So, for $y = |x - 4|+5$, the axis of symmetry is $x = 4$.

Step4: Identify the transformations

The parent function is $f(x)=|x|$. The transformation from $y = |x|$ to $y=|x - 4|+5$ is a horizontal shift of 4 units to the right (because of $x-4$) and a vertical shift of 5 units up (because of $+ 5$).

Answer:

Vertex: $(4,5)$
Axis of symmetry: $x = 4$
Transformations: 4 units right, 5 units up