Overview
This lecture explains how to transform the graphs of functions, focusing on the basic parabola f(x) = x². It covers vertical and horizontal shifts, vertical and horizontal stretches and shrinks, reflections, and how to combine these transformations. The same principles apply to other parent functions, such as |x|, x³, and √x.
Vertical Shifts
- Adding or subtracting a number outside the function (e.g., x² + 2 or x² - 3) moves the graph vertically.
- A positive value shifts the graph up; a negative value shifts it down.
- For x² + c, the vertex moves to (0, c).
- Example: x² + 2 is the graph of x² shifted up by 2 units; x² - 3 is shifted down by 3 units, with the vertex at (0, -3).
- Once you recognize this pattern, you can shift graphs without making a table of values.
Horizontal Shifts
- Adding or subtracting a number inside the function's input (e.g., (x + 2)² or (x - 2)²) shifts the graph horizontally.
- (x + a)² shifts the graph left by a units; (x - a)² shifts it right by a units.
- The vertex of (x + 2)² is at (-2, 0), showing a shift 2 units to the left.
- This can be less intuitive: a positive number inside the parentheses shifts the graph left, while a negative number shifts it right.
- Example: (x - 2)² moves the vertex to (2, 0), shifting the graph right by 2 units.
Vertical Stretches and Shrinks
- Multiplying the function by a constant (e.g., 2x² or (1/2)x²) changes the graph's vertical scale.
- If the coefficient is greater than 1, the graph becomes narrower (vertical stretch).
- If the coefficient is between 0 and 1, the graph becomes wider (vertical shrink).
- Example: 2x² is twice as narrow as x²; (1/2)x² is twice as wide.
- The graph grows faster or slower depending on the coefficient.
Horizontal Stretches and Shrinks
- Multiplying the x-variable inside the function (e.g., (2x)² or (1/2x)²) affects the graph horizontally.
- If the coefficient inside is greater than 1, the graph shrinks horizontally (becomes narrower).
- If the coefficient is less than 1, the graph stretches horizontally (becomes wider).
- This is the opposite of vertical stretches and shrinks: a larger coefficient compresses the graph horizontally, while a smaller one stretches it.
Reflections
- A negative sign outside the function (e.g., -x²) reflects the graph across the x-axis, flipping it upside down.
- A negative sign inside the function (e.g., (-x)²) reflects the graph across the y-axis.
- For x², reflecting across the y-axis does not change the graph, but for other functions like x³, it does.
- Example: -x² has the same shape as x² but opens downward.
Combined Transformations
- Multiple transformations can be applied in sequence to a function.
- Example: -2(x - 3)² + 4 involves:
- Vertical shift up by 4 units (vertex moves up)
- Horizontal shift right by 3 units (vertex moves right)
- Vertical stretch by a factor of 2 (graph becomes narrower)
- Reflection across the x-axis (graph opens downward)
- Apply each transformation step by step, starting from the basic function.
Generalization to Other Functions
- These transformation rules apply to many parent functions, not just x².
- For example, |x| - 2, x³ - 2, and √x - 2 are all shifted down by 2 units.
- Once you know the basic shape of a function, you can graph its transformations by applying these rules.
Key Terms & Definitions
- Vertical Shift: Moving the graph up or down by adding or subtracting a value outside the function.
- Horizontal Shift: Moving the graph left or right by adding or subtracting inside the function's input.
- Vertical Stretch/Shrink: Making the graph narrower or wider by multiplying the function by a constant.
- Horizontal Stretch/Shrink: Making the graph wider or narrower by multiplying the x inside the function.
- Reflection: Flipping the graph across the x-axis (negative outside) or y-axis (negative inside).
Action Items / Next Steps
- Review the summary table of transformations provided in the lecture for quick reference.
- Practice graphing functions with different combinations of transformations to reinforce understanding.
- Prepare for future lessons on more complex functions that cannot be reduced to simple transformations, which will require different graphing strategies.