Vertex Form of the Function
The translations you've learned about will move the graph up, down, left, or right. Sometimes you will see multiple translations in one problem.
The constant of the function will determine whether the graph will translate up or down. The number inside the absolute value symbol will determine whether the graph will translate left or right.
The vertex formula for translating the absolute value function is: f (x) = a|x - h| + k
Each variable in the function determines something about the translation of the absolute value function.
f (x) = a|x - h| + k
The value of a will determine whether the graph will open up or down.
The vertex of graph will be at (h, k). The value of h will be opposite of what is in the absolute value.
Opening of the Graph
The leading coefficient of the function will determine whether the graph opens up or down.
f (x) = a|x - h| + k
If your leading coefficient is positive, the graph will open up and your vertex will be a minimum point.
If a>0 (positive), the graph will open up.
If your leading coefficient is negative, the graph will open down and your vertex will be a maximum point.
If a>0 (negative), the graph will open down.
Example #5
Given the equation f (x) = |x - 3| + 4, determine the vertex of equation and whether the graph will open up or down.
The vertex of the equation is (3, 4). The vertex for this function will be a minimum point because the graph will open up and the leading coefficient If no leading coefficient is listed, assume it is 1. is greater than 0.
The graph of this equation will open up because a>0 (positive).
Example #6
Determine how the function f (x) = -2|x + 4| - 3 is translated from the parent graph f(x) = |x|.
The vertex of the equation is (-4, -3). This means the graph will be translated four units left and three units down. The vertex in this example will be a minimum point.
The graph will also be flipped over because the graph will open down. You know this because of the negative leading coefficient of the equation.
Example #7
Given the function f (x) = -|x - 2| - 5 determine the vertex, whether the vertex is a maximum or minimum point, and describe the translation.
The vertex of the graph will be at ( _____, _____ ).
The graph of the function will open: Up or Down
Will the vertex of the graph be a maximum or minimum point?
The graph is translated (choose one):
- Two units up and five units left
- Two units right and five units up
- Two units left and five units up
- Two units right and five units down
