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Math 150 – Fall 2015 Section 4C 1 of 6 Section 4C – Transformation of Functions Horizontal and Vertical Shifts Vertical Shift: Add a constant c to the function: y = f (x) + c. Equation: y = f (x) + c, c > 0 y = f (x) − c, c > 0 Transformation: Shift up c units Shift down c units Horizontal Shift Add a constant c to x before applying function: y = f (x + c). Equation: y = f (x + c), c > 0 y = f (x − c), c > 0 Transformation: Shift left c units Shift right c units Example 1. Transform the graph of f below to obtain the graph of the function. y = f (x) + 3 y = f (x − 2) y = f (x + 1) − 2 y = f (x − 1) + 2 Example 2. Apply vertical and horizontal shifts appropriately to the basic functions to obtain the graphs of the following. State the shift you applied. Determine the domain and range of each. (a) f (x) = (x − 2)2 − 3 Math 150 – Fall 2015 (b) f (x) = 1 x+2 Section 4C 2 of 6 +2 Reflections about the x- and y-axis Reflection about the x-axis: Equation: y = −f (x) Transformation: Reflect about the x-axis Reflection about the y-axis: Equation: y = f (−x) Transformation: Reflect about the y-axis Example 3. Transform the graph of f below to obtain the graph of the function. y = −f (x) y = f (−x) Example 4. Apply appropriate transformations to the graphs of basic functions to obtain the graphs of each of the following. List the basic function and the transformation. √ (a) y = −x Math 150 – Fall 2015 Section 4C 3 of 6 (b) y = −|x| Vertical Stretch and Shrink: Vertical Stretch and Shrink: Equation: y = cf (x), c > 1 Transformation: Stretch vertically by a factor of c Equation: y = cf (x), 0 < c < 1 Transformation: Shrink vertically by a factor of 1 c Example 5. Transform the graph of f below to obtain the graph of the function. y = 2f (x) y = 12 f (x) Example 6. Apply the appropriate transformations to the graphs of basic functions to obtain the graphs of each of the following. (a) y = 21 x3 Math 150 – Fall 2015 Section 4C 4 of 6 (b) y = 2|x| Combining Transformations Rules of Thumb: • Apply the reflections and vertical stretches or shrinks (put these in any order) before the horizontal and vertical shifts (put the shifts in either order). • To combine a horizontal shift with a reflection about the x-axis, factor out the negative first. For example, f (3 − x) should be re-written as f (−(x − 3)), with the transformations in order (1) Reflection about y-axis, (2) Shift right 3 units. • These rules are not the only way to combine transformations, but they will consistently give the correct answer. Example 7. Use transformations of basic functions to graph the following functions. State the basic function and each transformation in the order it was applied. For each function give (1) the domain and range, (2) intervals where y is increasing and decreasing, and (3) symmetry. (a) f (x) = − 21 (x + 2)2 − 4 √ (b) g(x) = 3 4 − x + 2 Math 150 – Fall 2015 Section 4C 5 of 6 Example 8. The following is the graph of y = f (x). Use transformation to graph the desired function. List any transformations and the order you performed them. y = f (x) y = 21 f (−x + 2) − 3 Example 9. The following graph is a transformation of a basic function. Find the basic function and all transformations in the order they were applied. Write the equation for this new function. Even and Odd Functions Definition. A function is even if and only if f (−x) = f (x), for all x in the domain of f . The graph of even function has symmetry about the y-axis. Even Function: for every x and −x, we have f (x) = f (−x). Math 150 – Fall 2015 Section 4C 6 of 6 Definition. A function f is odd if and only if f (−x) = −f (x) for every x in the domain of f . The graph of an odd function has symmetry about the origin. Odd Function: for every x and −x, we have f (x) = −f (−x). Testing whether a function is even, odd, or neither, evaluate f (−x): • If f (−x) = f (x), the function is even. • If f (−x) = −f (x), the function is odd. • If neither of the above is true, the function is neither even nor odd. Example 10. Determine algebraically whether the following functions are even, odd, or neither. (a) f (x) = 2|x2 −3| x4 +1 (b) f (x) = 3x4 − 4x3 − x−1 + 1 √ (c) f (x) = x x4 + 4 Example 11. Use the graphs to determine whether the functions are even, odd, or neither.