Question

lesson 14: absolute value functions (part 2)
cool down: elevations of places
the term \elevation\ is often used to describe the height of a place (such as a city, a mountain, or a valley) compared to sea level. for example, the city of houston, texas has an elevation of 105 feet. the surface of the sea has an elevation of 0 feet. some places are below sea level, so their elevations are negative values.

1. the table shows the elevation, ( e ), of several towns.

\begin{array}{|c|c|c|c|c|c|c|}hlinee & 180 & 12.1 & 5.4 & & -5.4 & -36 & -180 \hlinef(e) & & & & 0 & & & \hlineend{array}
function ( f ) gives the vertical distance of each town from sea level. both ( e ) and ( f(e) ) are measured in feet. complete the table of values.

1. write an equation to represent ( f(e) ).
2. two towns have different elevations, but when the elevations are used as inputs of ( f(e) ), they both produce an output of 25.

what are the elevations of the two towns? why do they produce the same output?

Explanation:

Step1: Calculate absolute values for table

For each elevation $e$, $f(e)=|e|$:

• $f(180)=|180|=180$
• $f(12.1)=|12.1|=12.1$
• $f(5.4)=|5.4|=5.4$
• $f(0)=|0|=0$
• $f(-5.4)=|-5.4|=5.4$
• $f(-36)=|-36|=36$
• $f(-180)=|-180|=180$

Step2: Define function for $f(e)$

Vertical distance from sea level is the absolute value of elevation, so $f(e)=|e|$.

Step3: Solve for elevations with $f(e)=25$

Set $|e|=25$. This gives two solutions:
$e=25$ or $e=-25$. The absolute value of both 25 and -25 is 25, meaning they are the same vertical distance from sea level (one above, one below).

Answer:

1. Completed table values for $f(e)$: 180, 12.1, 5.4, 0, 5.4, 36, 180
2. $f(e)=|e|$
3. The two elevations are 25 feet and -25 feet. They produce the same output because absolute value measures vertical distance from sea level, and both are 25 feet away from sea level (one above, one below).