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AP Calculus: Unit 1 (Pre-calculus)
Name_________________________
Quick Review:
1. Find the value of y that corresponds to x = 3 in y = -2 + 4(x – 3)
2. Find the value of m when x = -1 and y = -3.
2 y
m=
3 x
3. Determine whether the ordered pair is a solution to the equation.
3x – 4y = 5
(a) (2, ¼)
(b) (3, -1)
y = -2x + 5
(a) (-1, 7)
(b) (-2, 1)
4. Find the distance between the two points.
(a) (2, 1) and (1, -1/3)
(b) (1, 0) and (0, 1)
5. Solve for y in terms of x.
(a) 4x – 3y = 7
(b) -2x + 5y = -3
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Section 1.1: Lines
Learning Targets:
 I can write an equation and sketch a graph of a line given specific information.
 I can identify the relationships between parallel/perpendicular lines and slopes.
Example 1:
If a particle moves from the point (a, b) to the point (c, d), the slope would be:
Slopes
There are many ways to denote slope. Brainstorm some with your partner:
Parallel and Perpendicular Lines

The slopes of parallel lines are _____________________

The slopes of perpendicular lines are _____________________ (or the product of
the two slopes is _______)
Equations of Lines

Slope-intercept form:

Standard form (General Linear Equation):

Point-Slope form:
Example 1:
Write an equation for the line through the point (-1, 2) that is (a) parallel, and (b)
perpendicular to the line y = 3x – 4. (Leave your answers in point slope!!)
(a) Parallel:
(b) Perpendicular:
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Section 1.2 Notes: Functions and Graphs
Learning Targets:
 I can identify the domain and range of a function using its graph or equation.
 I can recognize even and odd functions using equations and graphs.
 I can interpret and find formulas for piecewise defined functions.
 I can write and evaluate compositions of two functions.
What is a function? Brainstorm with your partner!
Viewing and Interpreting Graphs
Example 2:
Identify the domain and range, and then sketch a graph of the function. No Calculator
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(A) y 
Parent function?________
(B) y  x  2  4
x 1
Parent function?________
Graph Viewing Skills
1. Recognize that the graph is reasonable.
2. See all important characteristics of the graph.
3. Interpret those characteristics.
4. Recognize grapher/calculator failure.
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Example 3:
Use an automatic grapher (calculator) to identify the domain and range, and then draw a
graph of the function.
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(A) y  4  x 2
(B) y  x 3
Even Functions and Odd Functions (Symmetry)


Even functions: f ( x)  f ( x) (Symmetric about the y-axis)
Odd functions: f ( x)   f ( x) (Rotation symmetric about the origin)
Example 4:
Identify the following functions as even or odd and explain why:
(A) y  2x 2
(B) y  ( x  1) 2
(C) y  x 5
(D) y  x 3  1
(E) y = cos x
(F) y = sin x
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Functions Defined in Pieces
Example 5:
 x , x  0

Graph y  f ( x)   x 2 ,0  x  1
1, x  1

Absolute Value Functions
Example 6:
Draw the graph of f ( x)  x  2  1. Then find the domain and range. (NO
CALCULATORS!)
Composite Functions
Example 7:
Find a formula for f ( g ( x)) if g ( x)  x 2 and f ( x)  x  7 . Then find f(g(2)) and g(f(2).
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Section 1.3 Notes: Exponential Functions
Learning Targets:
 I can determine the domain, range, and graph of an exponential function.
 I can solve problems involving exponential growth and decay.
 I can use exponential regression to solve problems.
Exponential Growth
Definition: Let a be a positive real number other than 1. The function f(x) = ax is the
exponential function with base a.
Example 1:
Graph the function y = 3(2x) – 4. State the domain and range.
Domain: ________________
Range: _________________
Example 2:
Find the zeros (solutions) of f(x) = (1/3)x – 4 graphically. (Sketch a picture of the
solution).
Zeros: ____________
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Rules for Exponents
If a > 0 and b > 0, the following hold for all real numbers x and y.
a x  b x  (ab) x
a x  a y  a x y
ax
 a x y
ay
(a x ) y  a xy
 ax 
a
    x 
b
b 
x
Exponential Decay
Definition: The half-life of a radioactive substance is the amount of time it takes for half
of the substance to change from its original radioactive state to a nonradioactive state by
emitting energy in the form of radiation.
Example 3:
Suppose the half-life of a certain radioactive substance is 20 days and that there are 5
grams present initially. When will there be only 1 gram of the substance remaining?
Definition: The function f(x) = kax , k > 0 is a model for exponential growth if a > 1,
and a model for exponential decay if 0 < a < 1.
The Number e
e  2.718281828
Example 4: Graph y = e x  2
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Example 5: Graph y = e 2 x
The number e is used in problems where interest (for example) is being compounded
continuously with the formula A(t) = Pert.
What is the formula that we use if we are not compounding continuously?
Example 6:
Chenelle opened a bank account at a 1.25% interest rate compounded quarterly. She put
$500 in the account 10 years ago and has not touched the account since then. How much
should be in her account today?
Example 7:
How long would it take Chenelle’s investment to double if the account was compounded
continuously?
To help you study for your quiz over 1.1-1.3, you may want to practice the quiz using the
“Quick Quiz” on page 29 of your textbook. These serve as a good review, but also great
AP testing practice!
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Parametric Relations: Activity
Objectives:
 Given parametric equations, plot relations by hand or calculator.
 Control the speed and direction of the plot by varying t and its increments or by
varying the equations.
 Produce parametric equations for Cartesian equations.
 Convert parametric equations to Cartesian equations (by eliminating the
parameter)
 Model motion problems.
Big Picture:
Parametrics offer a powerful method to plot many relations whether or not they are
functions. They also allow us to model motion, since we have more control over how
points are plotted. Beyond this introductory section, you will work with the calculus of
parametrics, so gaining a high level of comfort with them now will assure future success.
Activity:
When you first learned to plot lines, you probably used a chart where you chose x-values
and plugged them into an equation to produce y-values. With parametrics, the x- and yvalues are produced independently by substituting for a third variable, t, called a
parameter. In modeling motion, t usually represents time.
1. Given the following parametric equations, produce a table of values and plot the
relation. The table has been started for you.
xt  t 2  t
yt  2t
t
-2
-1
0
1
2
3
x
6
y
-4
2. Using substitution, convert the parametric equations in #1 to Cartesian form. This
is called eliminating the parameter. [Hint: Solve for x as a function of y.]
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Note: When parametric equations contain trig functions, we often rely on a trig identity
rather than substitution to eliminate the parameter. Consider the parametric equations:
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xt  sec t
y t  tan 2 t . If we use the trig identity 1  tan 2   sec 2  , the equation
2
becomes 1 + 2y = x2.
3. Convert to Cartesian coordinates: x = 3 sin(t), y = 4 cos(t). [Hint: Divide each by
the constant first. Then refer to the note above!]
Note: Any function can be converted to parametric form simply by letting the
independent variable be t. So, for instance, y  x 2  2 x can be converted to x = t,
y  t 2  2t . We must realize, though, that due to limitations on the values of t we will
not always produce a complete graph. For instance, y = 2x – 1 can be defined
parametrically as x = t, y = 2t – 1, but if t goes from [-10, 10], we would only see a plot of
a segment from (-10, -21) to (10, 19).
4. Determine parametric equations to plot the right half of the parabola y = (x – 2)2.
Graph it on your calculator to see if you have achieved your goal.
t min 
xt 
t max 
yt 
t step 
5. What is the effect of changing the increments of t or t step on the calculator? Find
out by exploring. Try the following examples, comparing A to B and A to C.
t min  0
t min  0
xt  2t  1
xt  2t  1
t max  3
t max  3
(A)
(B)
yt  t  1
yt  t  1
t step  0.2
t step  0.02
(C)
xt  2t  1
yt  t  1
t min  3
t max  0
t step  0.2
a. A compared to B: What was the effect of making the t step smaller? Explain
why it caused that effect.
b. A compared to C: What was the effect of a negative t step ? Explain why it
caused that effect.
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6. Compare the next two plots, where just the functions were changed slightly.
(Make sure you plot in radians.) Explain the similarities and differences in the
plots.
t min  0
t min  0
xt  3 cos t
xt  sin t
t max  6.3
t max  6.3
(D)
(E)
yt  3 sin t
yt  cos t
t step  0.1
t step  0.1
Section 1.4 Notes: Parametric Functions
Learning Targets:
 I can graph curves that are described using parametric equations.
 I can find parameterizations of circles, ellipses, line segments, and other curves.
Relations
Definition: A relation is a set of ordered pairs (x, y) of real numbers. The graph of a
relation is the set of points in the plane that correspond to the ordered pairs of the
relation. If x and y are functions of a third variable t, called a parameter, then we use the
parametric mode of our calculator.
Example 1:
x t
when t  0 . Indicate the
yt
direction in which the curve is being traced. Find a Cartesian equation for a curve that
contains the parametrized curve.
Describe the graph of the relation determined by
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Definition: If x and y are given as functions
x  f (t )
over an interval of t-values, then
y  g (t )
the set of points ( x, y )  ( f (t ), g (t )) defined by these equations is a parametric curve.
NOTE: If we are graphing a parametric curve on a closed interval [a, b], we consider the
point (f(a), g(a)) the initial point and (f(b), g(b)) the terminal point.
Example 2:
Describe the graph of the relation determined by x = 2 cos t,
y = 2 sin t, when 0  t  2 . Find the initial and terminal
points, if any, and indicate the direction in which the curve
is traced. Find a Cartesian equation for a curve that contains
the parametrized curve.
Ellipses
Example 3:
Graph the parametrized curve x = 3 cos t, y = 4 sin t,
0  t  2 . Find the Cartesian equation for a curve that
contains the parametric curve. Find the initial and terminal
points, if any, and indicate the direction in which the curve
is traced.
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Lines and Other Curves
Example 4:
Draw and identify the graph of the parametric curve
determined by x = 3t, y = 2 – 2t, 0  t  1.
Example 5:
Find a parametrization for the line segment with endpoints (-2, 1) and (3, 5).
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Section 1.5 Notes: Functions and Logarithms
Learning Targets:
 I can identify one to one functions.
 I can determine the algebraic representation and the graphical representation of a
function and its inverse.
 I can use parametric equations to graph inverse functions.
 I can apply the properties of logarithms.
 I can use logarithmic regression equations to solve problems.
One-to-one Functions
Definition: A function f(x) is one-to-one on a domain D if f(a)  f(b) whenever a  b.
NOTE: A one-to-one function passes the vertical line test AND the horizontal line test!
Example 1:
Determine if the following functions are one-to-one:
(A) f(x) = |x|
(B) g(x) =
x
Finding Inverses
Definition: The function defined by reversing a one-to-one function f is the inverse of f.
NOTE: If f ( g ( x))  g ( f ( x))  x , then f and g are inverses.
Writing f-1 as a function of x:
1. Solve the equation y = f(x) for x in terms of y.
2. Interchange x and y. The resulting formula will be y = f-1(x).
Example 2:
Show that the function y = f(x) = -2x + 4 is one-to-one and find its inverse function.
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Example 3:
(a) Graph the one-to-one function f(x) = x2, x  0 ,
together with its inverse and the line y = x, x  0 .
(b) Express the inverse of f as a function of x.
Logarithmic Functions
Definition: The base a logarithm function y  log a x is the inverse of the base a
exponential function y  a x ( a  0, a  1 ).
Properties of Logarithms
Example 4: Solve for x.
(B) e2x = 10
(A) ln x = 3t + 5
Properties of Logarithms:
For any real numbers x > 0 and y > 0,
1. Product Rule:
2. Quotient Rule:
3. Power Rule:
Example 5: Solve
a) ln(2x – 1) = ln 16
b) ln 56 – ln x = 4
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c) ln (x + 4) + ln x = ln 12
Definition: Change of Base Formula:
log a x 
ln x
ln a
Example 5:
Graph f ( x)  log 2 x .
Example 6:
Sarah invests $1000 in an account that earns 5.25% interest compounded annually. How
long will it take the account to reach $2500? (Solve algebraically and confirm
graphically!)
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Section 1.6 Notes: Trigonometric Functions
Learning Targets:
 I can convert between radians and degrees, and find arc length.
 I can identify the periodicity and even-odd properties of the trigonometric
functions.
 I can find values of trigonometric functions.
 I can generate the graphs of the trigonometric functions and explore different
transformations on these graphs.
 I can use the inverse trigonometric functions to solve problems.
Example 1:
Find all the trigonometric values of x if sin x = -3/5 and tan x < 0.
Transformations of Trigonometric Graphs
y  a cos( kx  c)  v
Example 2:
Determine the (a) period, (b) domain, (c) range,
and (d) draw the graph of the function
y  3 cos( 2 x   )  1
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Inverse Trig Functions:
Function
y  cos 1 x
Domain
Range
y  sin 1 x
y  tan 1 x
y  sec 1 x
y  csc 1 x
y  cot 1 x
Example 3:
Find the measure of cos-1 (-0.5) in degrees and radians (NO CALCULATOR!). How
many solutions should you expect to have?
Example 4:
Solve: (A) sin x = 0.7, 0  x  2
How many solutions should you
expect to have?
(B) tan x = -2,    x  
How many solutions should you
expect to have?
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