Tutorial
How does multiplying a function by \(-1\) (negating the function) affect the graph of this function?
- Click on any function button above to select it.
- Use the toggle switch to reflect the graph about the x-axis and observe the effect.
- Answer the following questions:
Question 1: What happens to points that are on the x-axis (where y = 0) when the graph is reflected about the x-axis?
Question 2: How does x-axis reflection affect the range of the function?
Question 3: What symmetry do you notice between the original and reflected graphs?
Question 4: For which functions does the reflected graph look "similar" to the original graph? (Hint: Try the sine function)
Explain analytically: For a function \(f(x)\), the transformed function \(-f(x)\):
- Reflects the graph about the x-axis (flips it upside down)
- Changes every point \((x, y)\) to \((x, -y)\)
- Negates all y-values (output values) of the function
- Changes the sign of the range: if original range is \([a, b]\), reflected range is \([-b, -a]\)
- Does not change the domain of the function
- Does not change x-intercepts (zeros of the function)
Key observations:
- X-axis reflection is equivalent to multiplying the function by \(-1\)
- The original and reflected graphs are symmetric with respect to the x-axis
- Points on the x-axis (where \(f(x) = 0\)) remain unchanged because \(0 = -0\)
- For odd functions (\(f(-x) = -f(x)\)), x-axis reflection is equivalent to y-axis reflection
- For even functions (\(f(-x) = f(x)\)), x-axis reflection creates a graph that is different from the original
Mathematical notation:
- Original: \(y = f(x)\)
- After x-axis reflection: \(y = -f(x)\)
- Equivalent transformation: \((x, y) \rightarrow (x, -y)\)