Title: Functions
1Functions
- Our objectives
- Recognize Parent Functions
- Graphically Algebraically
- Please take notes and ALWAYS ask questions ?
2Pre-Calculus 2
Todays Agenda
Do NOW Define domain and range in your
notebook. 10 mins
.
3The following basic graphs will be used
extensively in this section. It is important to
be able to sketch these from memory.
4 Constant Functionf(x) a
5 Linear function f(x) x
6 quadratic function
7 cubic function
8Polynomial Function
- http//zonalandeducation.com/mmts/functionInstitut
e/polynomialFunctions/graphs/polynomialFunctionGra
phs.html - zero degree
- first Degree
- second degree
- third degree
- Fourth degree
9Exponential Functionf(x) a
10Logarithmic Functionf(x)log x
11 square root function
12 cube root function
13absolute value function
14Rational Functionf(x)
15Reciprocal Functionf(x)
16Inverse Function
17Piece-wise Function
18Piece-wise Function
19We will now see how certain transformations
(operations) of a function change its graph. This
will give us a better idea of how to quickly
sketch the graph of certain functions. The
transformations are (1) translations, (2)
reflections, and (3) stretching.
20Vertical Translation
- Vertical Translation
- For b gt 0,
- the graph of y f(x) b is the graph of y
f(x) shifted up b units - the graph of y f(x) ? b is the graph of y
f(x) shifted down b units.
21Horizontal Translation
- Horizontal Translation
- For d gt 0,
- the graph of y f(x ? d) is the graph of y
f(x) shifted right d units - the graph of y f(x d) is the graph of y
f(x) shifted left d units.
22- Vertical shifts
- Moves the graph up or down
- Impacts only the y values of the function
- No changes are made to the x values
- Horizontal shifts
- Moves the graph left or right
- Impacts only the x values of the function
- No changes are made to the y values
23The values that translate the graph of a function
will occur as a number added or subtracted either
inside or outside a function.Numbers added or
subtracted inside translate left or right, while
numbers added or subtracted outside translate up
or down.
24Recognizing the shift from the equation, examples
of shifting the function f(x)
- Vertical shift of 3 units up
- Horizontal shift of 3 units left (HINT xs go
the opposite direction that you might believe.)
25Points represented by (x , y) on the graph of
f(x) become
If the point (6, -3) is on the graph of
f(x), find the corresponding point on the graph
of f(x3) 2
26Combining a vertical horizontal shift
- Example of function that is shifted down 4 units
and right 6 units from the original function.
27Reflections
- The graph of ?f(x) is the reflection of the
graph of f(x) across the x-axis. - The graph of f(?x) is the reflection of the
graph of f(x) across the y-axis. - If a point (x, y) is on the graph of f(x), then
- (x, ?y) is on the graph of ?f(x), and
- (?x, y) is on the graph of f(?x).
28Reflecting
- Across x-axis (y becomes negative, -f(x))
- Across y-axis (x becomes negative, f(-x))
29Vertical Stretching and Shrinking
- The graph of af(x) can be obtained from the
graph of f(x) by - stretching vertically for a gt 1, or
- shrinking vertically for 0 lt a lt 1.
- For a lt 0, the graph is also reflected across the
x-axis. - (The y-coordinates of the graph of y af(x) can
be obtained by multiplying the y-coordinates of y
f(x) by a.)
30VERTICAL STRETCH (SHRINK)
- ys do what we think they should If you see
3(f(x)), all ys are MULTIPLIED by 3 (its now 3
times as high or low!)
31Horizontal stretch shrink
- Were MULTIPLYING by an integer (not 1 or 0).
- xs do the opposite of what we think they should.
(If you see 3x in the equation where it used to
be an x, you DIVIDE all xs by 3, thus its
compressed horizontally.)