Question

Determine the Vertex of the following absolute value function.

State all translations

y =1/2|-X – 4|+5

describe how you break it down to the vertex in step by step form

Answer 1

vertex = (-4, 5)

Explanation:

In general, the graph of the absolute value function f(x) = a|x - h| + k will have its lowest value when f(x) = k (or highest value for f(x) = -a|x - h| + k). The lowest/highest value is the vertex (turning point).

Therefore, from inspection of the equation we can say that the y-coordinate of the vertex is 5.

Set the equation to 5 and then solve for x:

⇒ 5 = 1/2 |-X – 4| + 5

⇒ 1/2 |-X – 4| = 0

⇒ |-X – 4| = 0

Therefore, (-X - 4) = 0 and -(-X - 4) = 0

⇒ X = -4 (for both)

So the vertex is (-4, 5)

Translations

Absolute value parent function: f(x) = | -x |

Horizontal translation left 4 units: f(x) = |-x - 4|

Horizontal stretch of sf 1/2: f(x) = 1/2 |-x - 4|

Vertical translation up 5 units: f(x) = 1/2 |-x - 4| + 5