3. Real Numbers and Number Operations
• Whole numbers
• 0, 1, 2, 3 ……..
• Integers
• ….., -3, -2, -1, 0, 1, 2, 3 …….
• Rational numbers
• Numbers that can be written as the ratio of two integers.
• Irrational Numbers
• Real numbers that are not rational.
4. Equations
• Slope-intercept form of a linear equation y=mx + b
• Slope: m , y-intercept: b
• Standard form of a linear equation Ax + By = C
• A and B are not both zero.
12. Unit 4 Quadratic Functions (1)
• Terms:
• Quadratic function: form y=ax²+bx+c
• Parabola: u-shape
• Vertex: the lowest or the highest point
• Axis of symmetry: vertical line through the vertex
13. Unit 4 Quadratic Functions (2)
VERTEX
AXIS OF
SYMMETRY
PARABOLA
y=x²
14. How to graph PARABOLAS
1. Find the vertex
a) Use (𝑥 = −
𝑏
2𝑎
) to find the x-value.
b) Find the y-value of vertex (fill in x-value.)
2. Choose value for x and find y.
3. Plot the mirror image point from number 2.
4. Sketch the curve.
5. May need to repeat number 2 and 3.
15.
16.
17.
18. Factoring Patterns
1) Factoring Trinomials with Binomials
• x²+ bx + c = (x + m)(x + n)
= x²+ (m + n)x + mn
2) Factoring Difference of Squares
• a²- b²= (a + b)(a - b)
3) Factoring Perfect Square Trinomial
• a²+ 2ab + b²= (a + b) ²
• a²- 2ab + b²= (a - b) ²
19. MATH & HISTORY
• The First Telescope was made in 1608 by a German guy named Hans Lippershey.
• Refracting telescope: lenses magnify objects.
• Reflecting telescope: magnify objects with parabolic mirrors.
• Liquid telescope: made by spinning reflective liquids like mercury.
20. MATH & HISTORY (2)
Galileo first uses a
refracting telescope
for astronomical
purposes.
Isaac Newton
builds first
reflecting
telescope.
Maria
Mitchell is
first to use a
telescope to
discover a
comet.
Liquid mirrors are
first used to do
astronomical
research.
21. Fractals
• Definition
• A curve or geometric figure, each part of which has the
same statistical character as the whole.
• A geometric pattern that is repeated at every scale.
• An object or pattern that is "self-similar" at all scales.
26. nth Roots and Rational Exponents
• Let n be an integer greater than 1 and let a be a real number.
• If n is odd, then a has one real nth root:
• If n is even and a > 0 , then a has two real nth roots:
• If n is even and a = 0, then a has one nth root:
• If n is even and a < 0, then a has no real nth roots.
28. Properties of Exponents
Product of Powers Property
Power of A Power Property
Power of A Product Property
Negative Exponent Property
Zero Exponent Property
Quotient of Powers Property
Power of A Quotient Property
32. Inverse Functions
• An inverse relation maps the output values back to their
original input values.
• The domain of the inverse relation is the range of the original
relation and that the range of the inverse relation is the
domain of the original relation.
34. Exponential Growth
• An exponential function involves the expression b^x where
the base b is a positive number other than 1.
• ASYMPOTOTE: a line that a graph approaches as you move
away from the origin.
• If a > 0 , and b > 1 , the function ab^x is an exponential growth
function.
a = initial amount
r = percent decrease
1+r = growth factor
35. Compound Interest
• Consider an initial principal P deposited in an account that
pays interest at an annual rate r (expressed as a decimal),
compounded n times per year. The amount A in the account
after t years can be modeled by this equation:
37. The Number e
• Natural base e
• Euler number
• Discovered by Leonhard Euler (1707-1783)
• Natural base e is irrational.
• Defined as: As n approaches +, approaches
38. Logarithmic Functions
• Let b and y be positive numbers, b1. The logarithm of y with
base b is denoted by and defined as follows:
The expression is read as “log base b of y.”
40. Applications of Logarithm
• pH of solutions
• Decibels of sound
• Figuring interest
• Decaying radiation
• Short form for long numbers
• Signal decay
• Richter Scale – earthquakes
• F-stop in photography
• Oceanography
• Exponential growth
