An absolute value function with a vertex of (4,14) is negative on the interval (−∞,−2). On what other interval is the function negative?

Question

An absolute value function with a vertex of (4,14) is negative on the interval (−∞,−2).   On what other interval is the function negative?

Sadie Eaton · Accepted Answer

Step 1 The absolute value function with vertex (h,k) is defined as f(x)=a |x−h|+k. The vertex is (4,14). Therefore, f(x)=|x−4|+14. The function can be written without absolute value as f(x)={a(x−4)+14x&amp;gt;4−a(x−4)+14x&amp;lt;4 Step 2 The function is negative on the interval (−∞,−2). Since the function is continuous, the value of the function at x=−2 will be zero. Therefore, −a(−2−4)+14=0 −a(−6)+14=0 6a+14=0 6a=−14 a=−146 =−73 Step 3 Thus, the function becomes f(x)=−73|x−4|+14. Find the intervals on which f(x)=−73|x−4|+14 is negative as follows. −73|x−4|+14&amp;lt;0 −73|x−4|&amp;lt;−14 −73|x−4|&amp;gt;14 3⋅73|x−4|&amp;gt;14⋅3 7|x−4|&amp;gt;42 |x−4|&amp;gt;6 x−4&amp;lt;−6 or x−4&amp;gt;6 x&amp;lt;−6+4 or x&amp;gt;6+4 x&amp;lt;−2 or x&amp;gt;10 Step 4 Therefore, the function is negative on the intervalsx&amp;lt;−2 and x&amp;gt;10. That is, the function is negative on (−∞,−2)and(10,∞). Thus, the other interval in which the function negative is (10,∞).

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

An absolute value function with a vertex of (4,14) is negative on the interval (−oo,−2). On what other interval is the function negative?

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