Lesson 1-7: Transformations of Functions
See page 64 for basic parent functions.
What are transformations?
A function can be transformed by shifting right, left, up, or down. A function can also be transformed by a vertical or a horizontal stretch or compression. A function can be reflected across the x-axis or y-axis. These are all transformations.
If f (x) is a polynomial function:
Vertical Stretch or Compression
a f (x) | a | > 1 is a vertical stretch
| a | < 1 is a vertical compression
a < 0 is a reflection across the x-axis
Vertical Shifts
f (x) + k shift up k units
f (x) – k shift down k units
Horizontal Shifts
f (x – h) shift right h units
f (x + h) shift left h units
Reflections
– f (x) reflection across the x-axis
– f (–x) reflection across the y-axis
Horizontal Stretch or Compression
f (bx) horizontal stretch or compression of
| b | > 1 is a horizontal compression
| b | < 1 is a horizontal stretch
Summary:
g(x) = a· f (b(x – h )) + k
a is a vertical stretch or compression; if a < 0, g(x) reflects across the x-axis
b is a horizontal stretch or compression of
h is a horizontal shift of h units
k is a vertical shift of k units
3-4A: Symmetry 4/11/13
A function may be odd, even, or neither.
Odd Functions Even Functions
Origin Symmetry Y-Axis Symmetry
180º rotational symmetry Reflection across the Y-Axis
ALL odd exponents All even exponents
f (x) = sin x f (x) = cos x
g(x) = x3 g(x) = x4
h(x) = x5 + 2x3 – x h(x) = x8 – 5x6 + 3x4 – 7
f (x) = –f (–x) f (x) = f (–x)
( x , y ) and (–x , –y) are on the function. ( x , y ) and (–x , y) are on the function.
Lesson 3-7: Average Rate of Change
ARC is the slope of a secant line.
A secant line crosses a curve at two points.
Slope =
Difference Quotient =
To find the ARC using the difference quotient, let x = the x-value of the
point on the left and h = the horizontal distance between the two points.
Can Do Calculus: Instantaneous Rates of Change
IRC is the slope of the tangent line.
A tangent line touches a curve at one point.
Instantaneous Rate of Change (IRC) is the limit of the difference quotient as h approaches zero.
IRC is the derivative.
To find the IRC using the limit of the difference quotient:
How to find Standard Deviation:
(1) Find the mean (average).
(2) Find the deviations. Subtract the mean from each data value.
(3) Square the deviations.
(4) Average the squared deviations. This is the variance.
(5) Take the square root of the variance. This is the standard deviation.
Find the standard deviation for the following data: 4, 13, 5, 6, 9, 11
Inverse Functions
1. If f (x) passes the vertical line test, then it is a function.
2. If f (x) passes the horizontal line test, then its inverse is a function.
3. If f (x) and g(x) are inverses, then f (g(x)) = g(f (x)) = x.
f ○ g means f (g(x)) and g ○ f means g (f (x))
4. To find the inverse of a function, switch x and y then solve for y.
5. Graphs of inverse functions are reflections across the line y = x.
6. Symbolic notation for the inverse of f (x) is f –1(x).
7. If ( x , y ) is on the graph of f (x), then ( y , x ) is on the graph of f –1(x).
8. Slopes of inverse functions are reciprocals at mirror image points. In other words, the slope at (x , y) on f (x) is the reciprocal of the slope at (y , x) on f –1(x)