Section 1.6 & 1.7 Parent Functions and Transformations

Transcription

1 Math 150 c Lynch 1 of 8 Section 1.6 & 1.7 Parent Functions and Transformations Piecewise Functions Example 1. Graph the following piecewise functions. 2x + 3 if x < 0 (a) f(x) = x if x (b) f(x) = x x x < 1 x 1 (c) f(x) = x 3 + 1, x < 1 3, x = 1 x 2, 1 < x < 2 Example 2. For the previous graph in (c), find the following: (a) Domain: (b) Range: (c) x-intercept: (d) y-intercept: (e) f(0) = (f) f( 2) = (g) f( 1) =

2 Math 150 c Lynch 1.6 & 1.7 Functions 2 of 8 1. Constant Function: f(x) = c Catalog of Functions y-intercept: (0, c) Domain: (, ) x-intercept: none Range: {c} 2. Identity Function: f(x) = x y-intercept: (0, 0) Domain: (, ) x-intercept: (0, 0) Range: (, ) 3. Squaring Function: f(x) = x 2 y-intercept: (0, 0) Domain: (, ) x-intercept: (0, 0) Range: [0, ) 4. Cubing Function: f(x) = x 3 y-intercept: (0, 0) Domain: (, ) x-intercept: (0, 0) Range: (, ) 5. Square Root Function: f(x) = x y-intercept: (0, 0) Domain: [0, ) x-intercept: (0, 0) Range: [0, ) 6. Absolute Value Function: f(x) = x y-intercept: (0, 0) Domain: (, ) x-intercept: (0, 0) Range: [0, ) 7. Reciprocal Function: f(x) = 1 x y-intercept: none Domain: (, 0) (0, ) x-intercept: none Range: (, 0) (0, )

3 Math 150 c Lynch 1.6 & 1.7 Functions 3 of Transformations of Functions Horizontal and Vertical Shifts Vertical Shift: Add a constant c to the function: y = f(x) + c. y = f(x) + c, c > 0 y = f(x) c, c > 0 Shift up c units Shift down c units Horizontal Shift Add a constant c to x before applying function: y = f(x + c). y = f(x + c), c > 0 y = f(x c), c > 0 Shift left c units Shift right c units Example 3. 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 4. 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

4 Math 150 c Lynch 1.6 & 1.7 Functions 4 of 8 (b) f(x) = 1 x Reflection about the x-axis: y = f(x) Reflect about the x-axis Reflections about the x- and y-axis Reflection about the y-axis: y = f( x) Reflect about the y-axis Example 5. Transform the graph of f below to obtain the graph of the function. y = f(x) y = f( x)

5 Math 150 c Lynch 1.6 & 1.7 Functions 5 of 8 Example 6. 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 (b) y = x Vertical Stretch: Vertical Stretch and Shrink: y = cf(x), c > 1 Stretch vertically by a factor of c Vertical Shrink: y = cf(x), 0 < c < 1 Shrink vertically by a factor of 1 c Example 7. Transform the graph of f below to obtain the graph of the function. y = 2f(x) y = 1 2 f(x)

6 Math 150 c Lynch 1.6 & 1.7 Functions 6 of 8 Example 8. Apply the appropriate transformations to the graphs of basic functions to obtain the graphs of each of the following. (a) y = 1 2 x3 (b) y = 2 x Horizontal Stretch: Horizontal Stretch and Shrink y = f(cx), 0 < c < 1 Stretch horizontally by a factor of c Horizontal Shrink: y = f(cx), c > 1 Shrink horizontally by a factor of 1 c Example 9. Transform the graph of f below to obtain the graph of the function. y = f(2x) y = f( 1 2 x)

7 Math 150 c Lynch 1.6 & 1.7 Functions 7 of 8 Example 10. Apply the appropriate transformations to the graphs of basic functions to obtain the graphs of each of the following. (a) y = 2x (b) y = ( 1 3 x) 2 Combining Transformations Rules of Thumb: When combining multiple transformations, this is the order you should apply them to correctly graph your function. (This is not the only way to do it, but this method will give you the correct answer.) 1. Apply the horizontal shift first. 2. Apply the horizontal and vertical reflections, vertical stretches or shrinks, and horizontal stretches or shrinks next. You may put apply these in any order. 3. Apply the vertical shift last. Example 11. 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) = 1 2 (x + 2)2 4

8 Math 150 c Lynch 1.6 & 1.7 Functions 8 of 8 (b) g(x) = 4 2x + 2 Example 12. 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 = 1 2f( x + 2) 3 Example 13. 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.