Question

give two examples of functions that include an absolute value expression and have a vertex of (-1, 3).
select all that apply.
a. $f(x)=|x - 1| + 3$ b. $f(x)=|x + 1| - 3$
c. $f(x)=2|x - 1| - 3$ d. $f(x)=|x + 1| + 3$
e. $f(x)=|x - 1| - 3$ f. $f(x)=2|x - 1| + 3$
g. $f(x)=2|x + 1| - 3$ h. $f(x)=2|x + 1| + 3$

Explanation:

The general form of an absolute - value function with vertex \((h,k)\) is \(f(x)=a|x - h|+k\). Here, the vertex is \((- 1,3)\), so \(h=-1\) and \(k = 3\). The function should be in the form \(f(x)=a|x+1| + 3\), where \(a\) is a non - zero real number.

• For option A: \(f(x)=|x - 1|+3\) has vertex \((1,3)\), so it is incorrect.
• For option B: \(f(x)=|x + 1|-3\) has vertex \((-1,-3)\), so it is incorrect.
• For option C: \(f(x)=2|x - 1|-3\) has vertex \((1,-3)\), so it is incorrect.
• For option D: \(f(x)=|x + 1|+3\) has \(h=-1,k = 3\) (with \(a = 1\)), so it is correct.
• For option E: \(f(x)=|x - 1|-3\) has vertex \((1,-3)\), so it is incorrect.
• For option F: \(f(x)=2|x - 1|+3\) has vertex \((1,3)\), so it is incorrect.
• For option G: \(f(x)=2|x + 1|-3\) has vertex \((-1,-3)\), so it is incorrect.
• For option H: \(f(x)=2|x + 1|+3\) has \(h=-1,k = 3\) (with \(a = 2\)), so it is correct.

Answer:

D. \(f(x)=|x + 1|+3\), H. \(f(x)=2|x + 1|+3\)