linear, quadratic, cubic, power, trigonometric, exponential, and logarithmic functions are discussed.

1. Section 1.2
A Catalog of Essential Functions
V63.0121, Calculus I
January 22, 2009
Announcements
Blackboard is up
First HW due Thursday 1/29
ALEKS initial assessment due Friday 1/30

2. Outline
Modeling
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions

3. The Modeling Process
model
Real-world Mathematical
Problems Model
solve
test
interpret
Real-world Mathematical
Predictions Conclusions

4. Plato’s Cave

5. The Modeling Process
model
Real-world Mathematical
Problems Model
solve
test
interpret
Real-world Mathematical
Predictions Conclusions
Shadows Forms

6. Outline
Modeling
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions

7. Classes of Functions
linear functions, defined by slope an intercept, point and
point, or point and slope.
quadratic functions, cubic functions, power functions,
polynomials
rational functions
trigonometric functions
exponential/logarithmic functions

8. Linear functions
Linear functions have a constant rate of growth and are of the form
f (x) = mx + b.

9. Linear functions
Linear functions have a constant rate of growth and are of the form
f (x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile.
Write the fare f (x) as a function of distance x traveled.

10. Linear functions
Linear functions have a constant rate of growth and are of the form
f (x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile.
Write the fare f (x) as a function of distance x traveled.
Answer
If x is in miles and f (x) in dollars,
f (x) = 2.5 + 2x

11. Quadratic functions
These take the form
f (x) = ax 2 + bx + c

12. Quadratic functions
These take the form
f (x) = ax 2 + bx + c
The graph is a parabola which opens upward if a > 0, downward if
a < 0.

13. Cubic functions
These take the form
f (x) = ax 3 + bx 2 + cx + d

14. Other power functions
Whole number powers: f (x) = x n .
1
negative powers are reciprocals: x −3 = 3 .
x
√
fractional powers are roots: x 1/3 = 3 x.

15. Rational functions
Definition
A rational function is a quotient of polynomials.
Example
x 3 (x + 3)
The function f (x) = is rational.
(x + 2)(x − 1)

16. Trigonometric Functions
Sine and cosine
Tangent and cotangent
Secant and cosecant

17. Exponential and Logarithmic functions
exponential functions (for example f (x) = 2x )
logarithmic functions are their inverses (for example
f (x) = log2 (x))

18. Outline
Modeling
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions

19. Transformations of Functions
Take the sine function and graph these transformations:
π
sin x +
2
π
sin x −
2
π
sin (x) +
2
π
sin (x) −
2

20. Transformations of Functions
Take the sine function and graph these transformations:
π
sin x +
2
π
sin x −
2
π
sin (x) +
2
π
sin (x) −
2
Observe that if the fiddling occurs within the function, a
transformation is applied on the x-axis. After the function, to the
y -axis.

21. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f (x) + c, shift the graph of y = f (x) a distance c units
y = f (x) − c, shift the graph of y = f (x) a distance c units
y = f (x − c), shift the graph of y = f (x) a distance c units
y = f (x + c), shift the graph of y = f (x) a distance c units

22. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f (x) + c, shift the graph of y = f (x) a distance c units
upward
y = f (x) − c, shift the graph of y = f (x) a distance c units
y = f (x − c), shift the graph of y = f (x) a distance c units
y = f (x + c), shift the graph of y = f (x) a distance c units

23. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f (x) + c, shift the graph of y = f (x) a distance c units
upward
y = f (x) − c, shift the graph of y = f (x) a distance c units
downward
y = f (x − c), shift the graph of y = f (x) a distance c units
y = f (x + c), shift the graph of y = f (x) a distance c units

24. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f (x) + c, shift the graph of y = f (x) a distance c units
upward
y = f (x) − c, shift the graph of y = f (x) a distance c units
downward
y = f (x − c), shift the graph of y = f (x) a distance c units
to the right
y = f (x + c), shift the graph of y = f (x) a distance c units

25. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f (x) + c, shift the graph of y = f (x) a distance c units
upward
y = f (x) − c, shift the graph of y = f (x) a distance c units
downward
y = f (x − c), shift the graph of y = f (x) a distance c units
to the right
y = f (x + c), shift the graph of y = f (x) a distance c units
to the left

26. Outline
Modeling
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions

27. Composition is a compounding of functions in succession
g ◦f
g
x (g ◦ f )(x)
f
f (x)

28. Composing
Example
Let f (x) = x 2 and g (x) = sin x. Compute f g and g ◦ f .
◦

29. Composing
Example
Let f (x) = x 2 and g (x) = sin x. Compute f g and g ◦ f .
◦
Solution
f ◦ g (x) = sin2 x while g ◦ f (x) = sin(x 2 ). Note they are not the
same.

30. Decomposing
Example
Express x 2 − 4 as a composition of two functions. What is its
domain?
Solution √
We can write the expression as f ◦ g , where f (u) = u and
g (x) = x 2 − 4. The range of g needs to be within the domain of
f . To insure that x 2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.

31. The Far Side