Question

find the graph of an even function.

Explanation:

🆕 New Concept Discovered: Even and Odd Functions
Symmetric across the y-axis

Step 1: Define even function symmetry

An even function satisfies the algebraic condition:

$$ f(-x) = f(x) $$

Graphically, this means the graph is symmetric with respect to the \(y\)-axis. If you fold the graph along the vertical \(y\)-axis, the left and right halves will align perfectly (mirror images).

Step 2: Analyze the given graphs

• Top-Left Graph:
• Let's look at the \(y\)-intercept at \(x = 0\). The curve passes through \((0, 2)\).
• For \(x = \pi/2\), \(y = 0\). For \(x = -\pi/2\), \(y = 0\).
• For \(x = \pi\), the curve goes down towards negative infinity. For \(x = -\pi\), the curve goes up towards positive infinity.
• Since the left side does not mirror the right side, this graph is not symmetric about the \(y\)-axis.

• Top-Right Graph:
• The graph passes through the origin \((0,0)\).
• It has rotational symmetry about the origin (an odd function), not reflectional symmetry across the \(y\)-axis.

• Bottom-Left Graph:
• Let's check the points on either side of the \(y\)-axis:
• At \(x = 0\), \(y = -1\).
• At \(x = \pi/2\), \(y = 0\). At \(x = -\pi/2\), \(y = 0\).
• At \(x = \pi\), \(y = 1\). At \(x = -\pi\), \(y = 1\).
• The left half of the graph is a perfect mirror image of the right half across the vertical \(y\)-axis. This represents an even function (specifically, \(f(x) = -\cos(x)\)).

• Bottom-Right Graph:
• The graph passes through the origin \((0,0)\).
• It is symmetric about the origin (an odd function, specifically \(f(x) = \sin(x)\)), not across the \(y\)-axis.

Answer:

The graph of the even function is the bottom-left graph (which represents the function \(f(x) = -\cos(x)\)).