1. 1-6 R EFLECTIONS, ABSOLUTE VALUES, AND OTHER TRANSFORMATIONS Objective: Given a function, transformation it by reflection and by applying absolute value to the function or its argument.

2. R EFLECTIONS ACROSS THE X - AXIS AND Y - AXIS Graph:

3. C. Plot the pre-image, the reflections cross x -axis. d. Write an equation for the reflection. h(x) = - f(x) Graph:

4. Summary: Reflections Across the coordinate axes g(x) = f( - x) is a horizontal reflection of function f across the y -axis. h(x) = - f(x) is a vertical reflection of function f across the x -axis.

5. A BSOLUTE V ALUE TRANSFORMATIONS When you shoot basketball: While the basketball in the air, sometimes it is above (+ displacement) the basket level or below(-displacement). f(x)

6. However, the distance is the magnitude or size of the displacement, which is the absolute value of the displacement. k(x) = | f(x) | Graph k(x) f(x)

7. When you take the absolute value of the argument. p(x) = f ( | x |) f(x) if x ≥ 0 p(x) = f( - x) if x <0 Is p(x) a function ?

8. p(x) = f ( | x | ) When x ≥ 0 | x |= x, p(x ) = f (x) Ex : p (3) = f (3)=5 (3, 5) p (2)= f (2)=4 (2, 4) p (0) = f(0)=2 (0, 2) When x is negative | x |= - x, p(x) = f(-x) Ex : p (-3) = f (3)=5 (-3, 5) p (-2) = f (2)=4 (-2, 4) (3, 5) (2, 4) (0,2) f(x) (-2, -2) (-4, -1) (-3, 0)

9. Summary: The transformation g(x) = | f(x) | Reflects f across the x -axis if f(x) is negative. Leaves f unchanged if f(x) is nonnegative. The transformation g(x) = f(|x| ) Leaves f unchanged for nonnegative values of x. Eliminates the part of f for negative values of x. Reflects the part for positive values of x to the corresponding negative values of x.

10. E VEN AND ODD FUNCTIONS If f(x) = f( - x) for all x in the domain, then f(x) is an even function. If f(x) = - f( - x) for all x in the domain, then f(x) is an odd function.

12. Even: A function with only even exponents Odd: A function with only odd exponents. Even: reflection across the y-axis is the same as the pre-image. Ex6 : Is f(x) = x is an odd function? Sketch the graph. Odd: reflection across the y -axis gives the same image as reflection across the x -axis.