Shifting & reflecting functions | Algebra II | High School Math | Khan Academy | Summary and Q&A

TL;DR

The graphs of functions f(x) and g(x) have a relationship where g(x) is equal to f(x) plus 1, f(x) minus 2, or negative 1/3 times f(x).

Key Insights

• ✋ At the vertex of f(x), g(x) is exactly 1 higher than f(x).
• ❓ The distance between f(x) and g(x) vertically stays a constant 1.
• ❓ The relationship between f(x) and g(x) can be generalized as g(x) = f(x) ± 1 or g(x) = f(x) - 2.
• 😑 Shifting f(x) to the right or left can be achieved by adding or subtracting a constant in the function expression.
• 💱 The relationship between f(x) and g(x) changes when x is negative, with g(x) being 2 less than f(x).
• ❣️ Reflecting the graph of g(x) across the x-axis results in negative g(x), where each y-value becomes its opposite.
• 📈 The graph of g(x) can be obtained by multiplying negative g(x) by 3.

Transcript

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Questions & Answers

Q: What is the relationship between f(x) and g(x) in terms of adding or subtracting a constant?

The relationship between f(x) and g(x) is that g(x) is equal to f(x) plus 1 or f(x) minus 2. This means that g(x) is always 1 higher or 2 lower than f(x).

Q: How can the relationship between f(x) and g(x) be generalized for any x?

The relationship can be generalized as g(x) is always equal to f(x) plus 1, regardless of the value of x. This means that for any x, g(x) will be exactly 1 higher than f(x).

Q: How is the relationship between f(x) and g(x) different when x is negative?

When x is negative, the relationship between f(x) and g(x) is that g(x) is equal to f(x) minus 2. So, for any negative x, g(x) is always 2 less than f(x).

Q: What happens when the graph of g(x) is reflected across the x-axis?

When the graph of g(x) is reflected across the x-axis, it becomes negative g(x). This means that for any point (x, y) on g(x), the corresponding point on negative g(x) would be (x, -y).

Summary & Key Takeaways

• The graphs of f(x) and g(x) are related such that g(x) is equal to f(x) plus 1.

• Another relationship is that g(x) is equal to f(x) minus 2.

• There is a third relationship where g(x) is equal to negative 1/3 times f(x).