Y-axis Reflection of Graphs

Tutorial

How does replacing \(x\) with \(-x\) in a function affect the graph of this function?

1. Click on any function button above to select it.
2. Use the toggle switch to reflect the graph about the y-axis and observe the effect.
3. Answer the following questions:

Explain analytically: For a function \(f(x)\), the transformed function \(f(-x)\):

• Reflects the graph about the y-axis (flips it left-right)
• Changes every point \((x, y)\) to \((-x, y)\)
• Negates all x-values (input values) of the function
• Changes the domain: if original domain includes \(x\), reflected domain includes \(-x\)
• Does not change the range of the function
• Does not change y-intercepts (value at x=0)

Key observations:

• Y-axis reflection is equivalent to replacing \(x\) with \(-x\) in the function
• The original and reflected graphs are symmetric with respect to the y-axis
• Points on the y-axis (where \(x = 0\)) remain unchanged because \(f(0) = f(-0)\)
• For even functions (\(f(-x) = f(x)\)), y-axis reflection produces the same graph
• For odd functions (\(f(-x) = -f(x)\)), y-axis reflection is equivalent to x-axis reflection
• Functions that are neither even nor odd change appearance when reflected

Mathematical notation:

• Original: \(y = f(x)\)
• After y-axis reflection: \(y = f(-x)\)
• Equivalent transformation: \((x, y) \rightarrow (-x, y)\)

Function parity and y-axis reflection:

• Even functions: \(f(-x) = f(x)\) (symmetric about y-axis) - reflection doesn't change the graph
• Odd functions: \(f(-x) = -f(x)\) (symmetric about origin) - reflection flips both x and y values
• Neither even nor odd: Reflection changes the graph