MATH 117 Pre-calculus
Outline
• Part 1: Review of basic algebra and graphs,
and Straight lines (constant slope)
• Part 2: Functions, quadratic functions
(slope depends on x), and polynomials
• Part 3: Exponential functions (slope
depends on y) and logarithms
• Part 4: Oscillating functions: sine, cosine,
tangent; inverses; identities
Grading
• 4 tests (one over each part) each of which counts
as one grade
• Computer homework (8 assignments) the total of
which counts as one grade
• Regular written homework (9 assignments) the
total of which counts as one grade
• WebCT tests (7 assignments) which together count
as one grade.
• Final Exam which counts as three grades.
• Final Grade is determined by averaging the 10
grades.
Important Note
You must earn at least a “C” or better
grade in Math 117 to be eligible to move
on to Math 131 Calculus I at CBU.
Also, most institutions do not transfer courses
with grades less than “C”.
Usual notation
We will be using a lot of symbols in this course.
Sometimes we will use symbols based on the first
letter of a word, such as r for radius. Otherwise,
we will tend to use n and m for integers, x for the
independent variable, y for the dependent variable,
and the first letters of the alphabet for constants.
Also we will assume that the SQRT notation always
gives a positive. If we need to show that the
square root of 4 is both +2 and –2, we will write it
as ±SQRT(4) = ±√[4].
Basic Algebra
• Real numbers (on the number line)
– Rational numbers: r = n/m (where m is not 0,
and n and m are both integers)
– Irrational numbers: other numbers (for
example, π and e)
– Counter example: Square root of a negative
number is not a real number
Basic Algebra
• Decimal equivalence of fractions:
– Examples: 4/5 = 0.8 ; 4⅓ = 4.3333…
– Going backwards: d = .125125125…
now note that 1000*d = 125.125125…
now subtract: 1000*d – d = 125
so 999*d = 125, or d = 125/999
• Scientific notation:
– Examples: 1,500,000 = 1.5 x 106;
0.0025 = 2.5 x 10-3
• Calculators and approximations
Basic Algebra
• Interval notation:
– [3,5] means the interval that DOES include the
end points of 3 and 5.
– (3,5) means the interval that DOES NOT
include the end points of 3 and 5.
– Example: Newton’s Law of Gravity says F =
G*M*m/r2 which is good for (0,∞) since r is
the distance between two masses, M and m, and
r can’t be zero - it can only approach zero.
Basic Algebra
Absolute value: |x|
If x ≥ 0, then |x| = x; if x < 0, then |x| = -x.
Example of use: “distance”
|x - a| = d given a (starting location) &
d (distance, must be > 0), solve for the two
positions, x:
x = a + d, and x = a – d.
Example: |x - (-4)| = 7:
x = -4 + 7 = 3, and x = -4 - 7 = -11 .
Check: |3-(-4)| = 7 and |-11-(-4)| = 7 .
Basic Algebra
A direct relation is one where as one variable increases,
a related variable also increases. An example would
be y = 5x. A physical example is the pressure of the
water increases directly as the depth of the water
increases: P = ρgh (where P is the pressure and h is
the depth).
An inverse relation is one where as one variable
increases, a related variable decreases. An example
would be y = 5/x. A physical example is Newton’s
Law of Gravity: F = GMm/r2, as the distance from the
center of the earth, r, increases, the Force, F,
decreases as 1/r2. The force is inversely proportional
to the square of the radius.
Basic Algebra
Solving linear equations:
–
ax + b = c isolate the unknown (x)
divide every term by a: x + b/a = c/a;
now add –b/a to both sides: x = c/a – b/a .
– example: 5*x – 8 = -4
divide thru by 5: x – 8/5 = -4/5
add a negative –8/5 to both sides: x = -4/5 - -8/5
x = -4/5 + 8/5 = 4/5.
Check: 5(4/5) – 8 = -4 becomes 4 – 8 = -4.
Basic Algebra
Solving quadratic equations:
–
ax2 + bx + c = 0
write as: ax2 + bx = -c
divide thru by a: x2 + (b/a)x = -c/a
complete the square using fact that
(x+f)2 = x2 + 2fx + f2, so need to add (f)2 to each side where
here 2f = b/a: x2 + (b/a)x + (b/2a)2 = -c/a + (b/2a)2
so that we have [x+(b/2a)]2 = -c/a + b2/4a2
taking the square root and re-arranging gives:
x = [-(b/2a) ± SQRT{(b2/4a2) – (4ac/4a2)}
factoring out the 4a2 from the denominator of the square
root gives the standard form:
x = [-b ± SQRT{b2 – 4ac}] / 2a.
Basic Algebra
Besides linear and quadratic equations, we can keep
on upping the power of x. Equations of the form:
axn + bxn-1 + … + ux2 + vx + w = 0
are called polynomial equations of degree n.
Example: 17x4 + 3x3 – 8x2 – 9x = 11
is a polynomial of degree 4.
Usually these are solved graphically or numerically,
although sometimes we can use tricks to solve
some of the special cases. One special case is an
equation of the form: ax4 + bx2 + c = 0. Here we
simply use u=x2, to get au2 + bu + c = 0; solve for
u, and then solve for x.
Computer Homework
You should be able to do the first computer
homework program on Conversion Factors
(Volume #1, #1).
Note that when you work with applications, the
values usually have units that go with the values.
This means that instead of using abstract variables
like x and y, you will be using physical or
financial variables like pressure, temperature,
time, or money that have units.
Graphs
• The number line was a nice “picture” of
what we will call a 1-dimensional system.
• What about a plane? Can we identify a
position in a plane? What about your
address? Where is Memphis on a world
map?
• Cartesian (or rectangular) coordinates are
one way: (x,y)
Graphs
Points in a plane can be specified by their
coordinates: (x,y). For instance, a certain
point, A, will be specified by (xA, yA), and
another point, B, will be specified by (xB, yB).
y
10
5
-20
-10
10
-5
-10
20
30
x
Graphs
If point A is specified below, what are the
coordinates of point A?
If point B has coordinates of (-10,-15 ), where
would point B be located?
y
10
A
5
-20
-10
10
-5
-10
20
30
x
Graphs
The horizontal is usually specified as “x”, and
the vertical as “y”. This makes the coordinates
of point A (20,10) and point B (-10, -15).
y
10
A
5
-20
-10
10
-5
-10
B
20
30
x
Graphs
Can we define a “distance”, d, between points A
and B? How do we find (calculate) that
distance?
y
10
A
5
-20
-10
10
-5
-10
B
d
20
30
x
Graphs
By using the Pythagorean Theorem, we have:
d = SQRT[(xA-xB)2 + (yA-yB)2]
Does it matter if we use (xA-xB) or (xB-xA) ?
y
10
A
5
-20
-10
10
-5
-10
B
d
20
30
x
Graphs
For our example with A at (20,10) and
B at (-10,-15), we have
d = SQRT[(20-[-10])2 + (10-[-15])2] =
SQRT (302 + 252) = 39.05.
y
10
A
5
-20
-10
10
-5
-10
B
d
20
30
x
Graphs
Besides graphing points, we can graph equations
that relate y to x, since an equation is a relation
between values of y and values of x. This
graph, then, is a picture of the solutions of the
equation.
y
10
5
-20
-10
10
-5
-10
20
30
x
Graphs
The definition of a circle is that the points are
all equidistant from the center. We just
defined distance on a previous slide, so if
we start from (xc,yc) as the center point,
then the equation should be:
d = SQRT[(x-xc)2 + (y-yc)2], or
(x-xc)2 + (y-yc)2 = d2
Graphs
Example: Let’s choose the center point of the circle to
be (-10, 5), and the distance to be 15. Our equation
is then 15 = SQRT[(x-(-10))2 + (y-5)2], or
(x+10)2 + (y-5)2 = 152
d=15
y
10
5
-20
-10
10
-5
-10
20
30
x
Graphs
Let’s look at the graph of ax + by = c, and see if
we can notice anything special. To be
definite, let’s choose a = 4, b = -7, and c = -8.
The equation in this case is
4x – 7y = -8 .
Graphs
Example: 4x – 7y = -8 is an equation. If we
choose y=0, then the solution for x is x=-2. If
we choose y=5, then x=6.75. If we choose
y=15, then x=24.25. It looks like the points
fall on a straight line!
y
10
5
-20
-10
10
-5
-10
20
30
x
Graphs
Note that in both the graphs for the circle and
the straight line, we could see where the
circle or line cut across the x axis. This (or
these) is(are) the value(s) of x that satisfies the
equation when y=0. Thus, if we want to
solve an equation for x when y=0, we can
graph the equation and then look to see
what the value(s) of x is(are) when the
curve/line crosses the x axis!
Graphs
Example: From a previous slide, we plotted a circle
with the equation: (x+10)2 + (y-5)2 = 152 . If we set
y=0, we have the equation (x+10)2 + (-5)2 = 152 . By
looking at the graph, we can see that there are two
solutions, one near x=4 and the other near x=-24.
d=15
y
10
5
-20
-10
10
-5
-10
20
30
x
Graphs
Example: From a previous slide, we plotted a line
with the equation: 4x – 7y = -8 . If we set y=0, we
have the equation 4x – 0 = -8 . By looking at the
graph, we can see that there is only one solution, near
x=-2 .
y
10
5
-20
-10
10
-5
-10
20
30
x
Graphs
Notice that the circle equation,
(x-xc)2 + (y-yc)2 = d2 , was a second degree equation
and had the possibility of having two solutions.
Notice that the straight line equation,
4x – 7y = -8 , was a first degree equation and had the
possibility of having one solution.
Is the generalization that an n-th degree equation has
the possibility of having n solutions true?
Graphs
If we have an equation of the form:
ax + by = c, can you recognize this as a straight
line?
If we have an equation of the form:
x2 + bx + y2 + dy = f, can you recognize this as
being any special type of line? Could you put it in
“standard form” for that special type? [Hint:
complete the square for both x and y.]
Graphs
Example: Given x2 + 3x + y2 - 4y = 11, put this in the
form of a circle.
First we complete the square for the x:
x2 + 3x + (3/2)2 + y2 - 4y = 11 + (3/2)2 , or
(x+1.5)2 + y2 - 4y = 13.25. Now we complete the
square for the y:
(x+1.5)2 + y2 - 4y + (4/2)2 = 13.25 + (4/2)2 , or
(x+1.5)2 + (y-2)2 = 17.25 where we can identify
xc = -1.5, yc = 2, and d = SQRT(17.25) .
Graphs
What would happen if we multiplied the x2
term by a constant other than 1?
What would happen if we multiplied the y2
term by a constant other than 1?
What would happen if either of those
constants were negative?
What would happen if both of those constants
were negative?
Graphs
Example: What about the graph of absolute
value: |x - a| = d ? Can you “picture” what
this will look like? What do the constants a
and d do for the graph?
Graphs
Given: y = |x - (-4)| (here d = y, and a = -4)
when x = 0, y = 4
when x = 2, y = 6
y
when x = 4, y = 8
8
when x = -2, y = 2
when x = -4, y = 0
-8
8
when x = -6, y = 2
when x = -8, y = 4
-8
x
Regular Homework set #1
(continued on next page)
1. Find the fractional expression (s/r) for the
rational number 3.153846153846153846…
(where the 153846 sequence continues to repeat)
For problems 2-7, solve all for x, and then check
your answer:
2. |x + 6| = 12
3. -3x + 4 = -2
4. 5x2 + 4x + 7 = 0
5. You choose non-zero values for a, b, and c; and
solve ax2 + bx + c = 0 for x, and check your
answer.
Regular Homework Set #1
(Continued from previous page)
-4x2 + 7x – 3 = 8
[3/(x+5)] = [4/(x-7)] Hint: get a common
denominator, and as long as the denominator is
not zero, you only have to make the numerator
zero.
8. Given the equation x2 + 9x + y2 – 4y = 20, find
the radius and the center of the circle.
9. Given the center of a circle is at (2, -7), and the
circle has a radius of 12, determine the equation
relating x and y.
10. Where (what values of x) does this circle of #9
cross the x axis?
6.
7.
Computer Homework
You should be able to do the second computer
homework program Relations (Volume 0, #1).
Note that when you graph data in applications, the
values usually have units that go with the values.
This means that instead of using abstract variables
like x and y, you will be using physical or
financial variables like pressure, temperature,
time, or money that have units. That means that
the axes for graphs will also have units with them.
Graphing Calculators
We’ll now spend time with the graphing calculators.
Diamond / F1 (Y=)
Use up and down areas and CLEAR button to clear out
any old functions
Type in: 2x+3 then press diamond / F3 (graph)
Diamond / F2 (Window) shows min & max x and y
and scales used. Adjust these to see area of interest.
Note that x is about twice as long as y is high on
display.
Graphing Calculators
In looking at graphs of relations, we can look for
different kinds of symmetries (samenesses).
Does the graph look the same on the left side of the y
axis as it does on the right side (mirror reflection);
if so (x,y) = (-x,y).
Does the graph look the same above the horizontal
(x) axis as it does below (mirror reflection);
if so (x,y) = (x,-y).
Does the graph look the same through the origin; if
so (x,y) = (-x,-y)
Regular Homework set #2
1. Graph the equation: y = x3 – 5x +2x +4, on your
calculator, find the local max and min by reading
your graph, and find by reading the graph the
values of x where the curve crosses the x axis
(where y=0).
2. Graph the two equations: y = 2x4 – 3x2 + 7 and
y = x3 – 5x + 2x + 4. Make a sketch of the plot
from the calculator. By reading your graph, find
all approximate values of x where the two
curves intersect.