This document provides an outline and overview of essential functions and concepts covered in a Calculus I course. It includes sections on modeling, classes of functions (linear, polynomial, power, rational, trigonometric, exponential, logarithmic), transformations of functions, and vertical and horizontal shifts of functions. Examples are provided to illustrate key concepts like writing linear equations to model real-world situations, finding the equation of a parabola through three points, and how transformations affect a function graph.

1. Section 1.2
A catalog of essential functions
V63.0121.006/016, Calculus I
January 21, 2010
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. . . . . .

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

3. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
. . . . . .

4. Plato’s Cave
. . . . . .

5. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
S
. hadows F
. orms
. . . . . .

6. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial 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. Example
Biologists have noticed that the chirping rate of crickets of a
certain species is related to temperature, and the relationship
appears to be very nearly linear. A cricket produces 113 chirps
per minute at 70 ◦ F and 173 chirps per minute at 80 ◦ F.
(a) Write a linear equation that models the temperature T as a
function of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate
the temperature.
. . . . . .

12. Solution
. . . . . .

13. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
. . . . . .

14. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
. . . . . .

15. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
. . . . . .

16. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
37
If N = 150, then T = + 70 = 76 1 ◦ F
6
6
. . . . . .

17. Other Polynomial functions
Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0,
downward if a < 0.
. . . . . .

18. Other Polynomial functions
Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0,
downward if a < 0.
Cubic functions take the form
f(x) = ax3 + bx2 + cx + d
. . . . . .

19. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
. . . . . .

20. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c.
. . . . . .

21. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:
3 = a · 02 + b · 0 + c
−1 = a · 2 2 + b · 2 + c
0 = a · 32 + b · 3 + c
. . . . . .

22. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:
3 = a · 02 + b · 0 + c
−1 = a · 2 2 + b · 2 + c
0 = a · 32 + b · 3 + c
Right away we see c = 3. The other two equations become
−4 = 4a + 2b
−3 = 9a + 3b
. . . . . .

23. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
. . . . . .

24. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
Subtract these two and we have −6 = −6a =⇒ a = 1.
. . . . . .

25. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
Subtract these two and we have −6 = −6a =⇒ a = 1.
Substitute a = 1 into the first equation and we have
−12 = 12 + 6b =⇒ b = −4
. . . . . .

26. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
Subtract these two and we have −6 = −6a =⇒ a = 1.
Substitute a = 1 into the first equation and we have
−12 = 12 + 6b =⇒ b = −4
So our equation is
y = x2 − 4x + 3
. . . . . .

27. Other power functions
Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.
. . . . . .

28. 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)
. . . . . .

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

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

31. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .

32. 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
. . . . . .

33. 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.
. . . . . .

34. 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
. . . . . .

35. 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
. . . . . .

36. 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
. . . . . .

37. 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
. . . . . .

38. 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
. . . . . .

39. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .

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

41. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
. . . . . .

42. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the
same.
. . . . . .

43. Decomposing
Example
√
Express x2 − 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) = x2 − 4. The range of g needs to be within the domain of f.
To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
. . . . . .