Problem 102 - College Algebra with Modeling and Visualization

Question

If an even function \(f\) has an absolute minimum of \(-6\) at \(x=-2,\) then what else can be said about \(f ?\) Explain.

Step-by-Step Solution

Answer

The function has an absolute minimum of \(-6\) at \(x = 2\) as well.

Step 1: Understand Even Functions

An even function is a function where for every value of \(x\), \(f(x) = f(-x)\). This means the function is symmetric about the y-axis.

Step 2: Determine Absolute Minimum

Given that an absolute minimum of \(-6\) occurs at \(x = -2\), we know \(f(-2) = -6\). Since this is an absolute minimum, it is the lowest value the function takes on its entire domain.

Step 3: Use Symmetry Property

Because \(f\) is an even function, the symmetry property implies that \(f(2) = f(-2)\). Hence, \(f(2) = -6\).

Step 4: Conclusion About the Minimum

Since the function is even and has an absolute minimum at \(x = -2\), it also has an absolute minimum at \(x = 2\). Thus, the function \(f(x)\) has an absolute minimum of \(-6\) at both \(x = 2\) and \(x = -2\).

Key Concepts

Absolute Minimum

When we talk about the absolute minimum of a function, it refers to the smallest value that the function can take over its entire domain. This value is significant because it represents the lowest point on the graph of the function. Imagine the graph as a mountain range - the absolute minimum would be the deepest valley. For example, if the absolute minimum is at , it means that for all values of the function across its entire input range, no value is lower than . Several characteristics should be remembered about absolute minimums:

• It can occur at more than one point.
• It is not possible for the function’s values below this point.
• An absolute minimum can be at the endpoints of the domain or anywhere in between.

For an even function, once an absolute minimum is found at a certain point, due to symmetry, it typically mirrors on the opposite side of the y-axis.

Function Symmetry

Symmetry in functions is a fascinating concept where the function can exhibit repetitive patterns. An even function is a type, specifically one that shows symmetry about the y-axis. This means for every input, the function values at positive and negative inputs are the same. Mathematically, we express this as . Why is this symmetry important?

• It simplifies analysis - we only need to study one part of the graph.
• Many real-world systems are modeled using symmetric functions.
• Even functions help determine certain properties, such as maxima and minima, based on one side of the graph.

In the context of our function with an absolute minimum of at , function symmetry tells us that will also have this minimum because it mimics both sides of the y-axis.

Y-axis Symmetry

Y-axis symmetry is a specific kind of symmetry where the left and right sides of a graph reflect over the y-axis. When a function has this property, it means that for every point on one side, there is a matching point on the other side. This symmetry simplifies many problems because it reduces the amount of analysis needed - if you know what happens on one side, you automatically know what happens on the other. Why is this symmetry crucial?

• It makes sketching graphs easier – you only need to draw half and then reflect.
• In calculus, it can simplify integrations and derivations.
• Key characteristics, like peaks and valleys, appear symmetrically, providing predictable behavior.

Considering our even function, having y-axis symmetry directly tells us that if the function reaches an absolute minimum at one point like , it does so symmetrically at as well, ensuring balance in the graph's layout.