1. Unit 6 Lesson 8

2. Do Now Find the following values for f(x) and g(x): f(x) = x 2 g(x) = 2x 2 f(1); f(2); f(0); f(-1); f(-2) g(1); g(2); g(0); g(-1); g(-2)

3. HW & Objectives Be able to reflect graphs in the x and y axes Be able to shrink or stretch graphs Be able to recognize odd and even functions HW: Read p. 249 Do p. 251: 6, 9, 12-28 even, 36, 38, 49, 51, 53, 65, 67, 69

4. Review To shift any function, f(x), 3 units up, how does the equation of the function change? Add 3: f(x) + 3 To shift any function, f(x), 4 units to the left, how does the equation of the function change? Replace x with x + 4: f(x + 4)

5. Review If f(x) = x 2, then vertex is at (0, 0). Where would the vertex be for: a) g(x) = (x + 5) 2 – 6 (-5, -6) b) h(x) = 2(x – 1) 2 + 8 (1, 8)

6. New: Reflecting in x and y axes Go to fooplot and graph f(x) = x^2 and –x^2; how do the graphs compare? Do the same with g(x) = 4x^2 and -4x^2; how do the graphs compare? The graphs are the same except one opens up and one opens down. They are the same except they are reflected over the x-axis f(x) and –f(x) are the same graphs reflected in the x-axis

7. New: Reflecting in x and y axes Reflecting in the y-axis occurs when x is replaced in a function with –x For example, on fooplot, graph f(x) = x^3 and g(x) = (-x)^3 How do the graphs compare? Try (2x)^(1/2) and (-2x)^(1/2) How do the graphs compare?

8. New: Stretching and Shrinking Using fooplot, compare: f(x) = x^3; g(x) = 5(x^3); and 1/5(x^3) These are examples of vertical stretching and shrinking While nothing changes with the x-values, all the f(x) values are changed by a factor of 5 (stretch) or a factor of 1/5 (shrink)

9. New: Stretching and Shrinking To stretch or shrink f(x) = x^3 horizontally, replace x with (5x) to shrink it, and (1/5)x to stretch it Try this on fooplot; compare to the graphs from the previous slide also Notice that replacing x with 5x means point (1,1) is transformed to (1/5, 1) Notice that replacing x with (1/5)x means point (1,1) is transformed to (5,1)

10. All Together: Vertical and Horizontal shift, stretch, and shrink; x & y axes reflections Vertical Shift: add cf(x) + cc units up Horizontal Shift: replace x with x + cf(x + c)c units left Vertical Stretch/Shrink: multiply by ccf(x)c > 1, stretch 0 < c < 1, shrink Horizontal Stretch/Shrink: replace x with cxf(cx)c > 1, shrink by 1/c 0 < c < 1, stretch by 1/c Reflect over x-axis:Reflect over y-axis f(x)  -f(x)f(x)  f(-x)

11. Practice Start with f(x) = x^2; point (1,1) is on the graph Stretch the graph vertically by a factor of 3; (1,1) corresponds to: f(x) = 3(x^2); (1,3)  vertical coordinate changes Shrink vertically by 1/3; (1,1) corresponds to: f(x) = (1/3)(x^2); (1, 1/3)  vert. coord. changes Shrink horizontally by 3: f(x) = (3x)^2; (1/3,1) is on the graph  horizontal coordinate changes Stretch horizontally by 1/3: f(x) = ((1/3)x)^2; (9, 1) on graph  hor. coord. changes

12. New: Even and Odd Functions f(x) = x^2 is an example of an even function Even functions are symmetrical with respect to the y axis; also, f(x) = f(-x) f(x) = x^3 is an example of an odd function Odd functions are symmetrical around the origin