1.4 Shifts, Reflections, and Stretches

1. 1.4 Shifts, Reflections, and Stretches

2. 6 Common parent functions • Constant linear absolute value • Quadratic cubic square root

3. Vertical and horizontal shifts • The following changes in the functions y=f(x) will produce the stated shifts in the graph of y=f(x): • H(x)=f(x-c) horizontal shift c units to the right • H(x)=f(x+c) horizontal shift c units to the left • H(x)=f(x)-c vertical shift c units downward • H(x)=f(x)+c vertical shift c units upward

4. Describing shifts • Describe the shifts of generated by the following

5. Answer • Vertical shift down one unit • Horizontal shift right one unit • Horizontal shift left two units and vertical shift up one unit

6. Reflections • The following changes in the function y=f(x) will produce the stated reflections in the graph of y=f(x): • H(x)=-f(x) reflection with respect to the x-axis • H(x)=f(-x) reflection with respect to the y-axis

7. Describing reflections • Describe the reflections of generated by the following

8. Answer • Therefore reflected with respect to the y-axis • Therefore reflected with respect to the x-axis

9. Nonrigid transformations • Nonrigid transformations actually distort the shape of the graph, instead of just shifting or reflecting it. • Nonrigid transformations of y=f(x) come from equations of the form y=cf(x). • If c>1, then there is a vertical stretch of the graph of y=f(x). • If 0<c<1, then there is a vertical shrink • We will discuss horizontal extensions and compressions at a further date.

10. Describing stretches • Describe the stretches of generated by the following

11. Answer • Because 3 factored out and meant the graph was stretched by 3 in the vertical direction

12. Put it all together • Determine all shifts, reflections, and stretches of the absolute value parent function given by the following

13. Answer • - (reflect over x-axis) • 5 (stretch vertically by 5) • +2 (shift horizontally left 2 units) • -6 (shift vertically down 6 units)