Pre-Calculus
1) Functions and their Graphs:
· Properties of Lines
1. Straight Line: a line that never changes direction
2. Slope intercept form: the equation of a straight line in the form y = mx + b where m is the slope of the line and b is its y-intercept
3. X-intercept: is the point where a line, parabola, or ellipses touches the x-axis
4. Y-intercept: is the point where a line, parabola, or ellipses touches the y-axis
· Properties of Lines
1. Straight Line: a line that never changes direction
2. Slope intercept form: the equation of a straight line in the form y = mx + b where m is the slope of the line and b is its y-intercept
3. X-intercept: is the point where a line, parabola, or ellipses touches the x-axis
4. Y-intercept: is the point where a line, parabola, or ellipses touches the y-axis
· Basic Functions, Functions and Graphs
1. Identity Function: f(x)=x 7. Exponential Growth Function: f(x)= ex
2. Squaring Function: f(x)= x^2 8. Basic Logistic Function: f(x)= 1/1+e-x
3. Cubing Function: f(x)= x^3 9. Natural Logarithmic Function: f(x)= ln x
4. Abosulte Value Function: f(x)= │x│=abs(x) 10. Greatest Integer Function: int(x)
5. Reciprocal Function: f(x)= 1/x 11. Sine Function: f(x)= sin(x)
6. Square Function: f(x)= √x 12. Cosine Function: f(x)= cos(x)
2. Squaring Function: f(x)= x^2 8. Basic Logistic Function: f(x)= 1/1+e-x
3. Cubing Function: f(x)= x^3 9. Natural Logarithmic Function: f(x)= ln x
4. Abosulte Value Function: f(x)= │x│=abs(x) 10. Greatest Integer Function: int(x)
5. Reciprocal Function: f(x)= 1/x 11. Sine Function: f(x)= sin(x)
6. Square Function: f(x)= √x 12. Cosine Function: f(x)= cos(x)
Transformations
1. Shifts: They are translations.
2. Translations: A translation is an operation that shifts a graph horizontally, vertically, or both.
Horizontal Translation: Vertical Translation:
y = f (x - c) a translation to the right by c units y = f (x) + c a translation up by c units
y = f (x + c) a translation to the left by c units y =f (x) - c a translation down by c units
3. Stretches/Shrink: A stretch/shrink is when a plane figure distorted vertically of horizontal.
Horizontal Stretch/Shrink: Vertically Stretch/Shrink:
y = f (x/c) a stretch by a factor of c if c > 1 y = c * f (x) a stretch by a factor of c if c > 1
y = f (x/c) a stretch by a factor of c if c < 1 y = c * f (x) a stretch by a factor of c if c < 1
4. Reflection: is a transformation in which the figure is the mirror image of the other.
x-axis: y = -f (x) y-axis: y = f (-x)
2. Translations: A translation is an operation that shifts a graph horizontally, vertically, or both.
Horizontal Translation: Vertical Translation:
y = f (x - c) a translation to the right by c units y = f (x) + c a translation up by c units
y = f (x + c) a translation to the left by c units y =f (x) - c a translation down by c units
3. Stretches/Shrink: A stretch/shrink is when a plane figure distorted vertically of horizontal.
Horizontal Stretch/Shrink: Vertically Stretch/Shrink:
y = f (x/c) a stretch by a factor of c if c > 1 y = c * f (x) a stretch by a factor of c if c > 1
y = f (x/c) a stretch by a factor of c if c < 1 y = c * f (x) a stretch by a factor of c if c < 1
4. Reflection: is a transformation in which the figure is the mirror image of the other.
x-axis: y = -f (x) y-axis: y = f (-x)
Combining Functions
1. Sum: (f + g)(x) = f (x) + g (x)
2. Difference: (f - g)(x) = f (x) - g (x)
3. Product: (f g)(x) = f (x)g(x)
4. Quotient: (f/g)(x) = f(x)/ g (x), Sum: (f + g)(x) = f (x) + g (x)
5. Difference: (f - g)(x) = f (x) - g (x)
6. Product: (f g)(x) = f (x)g(x)
7. Quotient: (f/g)(x) = f(x)/ g (x)
2. Difference: (f - g)(x) = f (x) - g (x)
3. Product: (f g)(x) = f (x)g(x)
4. Quotient: (f/g)(x) = f(x)/ g (x), Sum: (f + g)(x) = f (x) + g (x)
5. Difference: (f - g)(x) = f (x) - g (x)
6. Product: (f g)(x) = f (x)g(x)
7. Quotient: (f/g)(x) = f(x)/ g (x)
2) Polynomials and Rational Functions
1. Quadratic Function: is a function that can be described by an equation of the form f(x) = ax^2 + bx + c, where a ≠0.
2. Polynomial: can have constants, variables and exponents, but never division by a variable.
Ex: 4xy^2+3x-5
3. Synthetic Division: A method of dividing polynomials when the divisor is a polynomial of the first degree, by using only the coefficients of the terms.
4. Complex number: a combination of a real number and an imaginary number.
Example: (3 + 2i)(1 + 7i) (3 + 2i)(1 + 7i)
= 3×1 + 3×7i + 2i×1+ 2i×7i
= 3 + 21i + 2i + 14i2
= 3 + 21i + 2i - 14
(because i2 = -1)
= -11 + 23i
5. Rational Functions: is any function which can be written as the ratio of two polynomial functions.
Ex: x^2 +5/ x^2+10
6. The fundamental theorem of algebra: states that every polynomial equation over the field of complex numbers of degree higher than one has a complex solution.
2. Polynomial: can have constants, variables and exponents, but never division by a variable.
Ex: 4xy^2+3x-5
3. Synthetic Division: A method of dividing polynomials when the divisor is a polynomial of the first degree, by using only the coefficients of the terms.
4. Complex number: a combination of a real number and an imaginary number.
Example: (3 + 2i)(1 + 7i) (3 + 2i)(1 + 7i)
= 3×1 + 3×7i + 2i×1+ 2i×7i
= 3 + 21i + 2i + 14i2
= 3 + 21i + 2i - 14
(because i2 = -1)
= -11 + 23i
5. Rational Functions: is any function which can be written as the ratio of two polynomial functions.
Ex: x^2 +5/ x^2+10
6. The fundamental theorem of algebra: states that every polynomial equation over the field of complex numbers of degree higher than one has a complex solution.
3) Exponential and Logarithm Functions
1. Exponential Functions: the exponential function f with base is denoted by f(x)=ax, where a > 0, a≠1, and x is any real number.
Ex#1: a. f(x) =2^x b. f(x)= 2^-x c. f(x)= 0.6^x
Ex #2: The graph of y=2x is shown to the right. Here are some properties of the exponential function when the base is greater than 1.
Ex#1: a. f(x) =2^x b. f(x)= 2^-x c. f(x)= 0.6^x
Ex #2: The graph of y=2x is shown to the right. Here are some properties of the exponential function when the base is greater than 1.
- The graph passes through the point (0,1)
- The domain is all real numbers
- The range is y>0.
- The graph is increasing
- The graph is asymptotic to the x-axis as x approaches negative infinity
- The graph increases without bound as x approaches positive infinity
- The graph is continuous
- The graph is smooth
2. Logarithm Functions: logarithmic functions are just the inverse of exponential functions
Ex:
1. Properties of logarithms: 1. loga (uv) = loga u + loga v 1. ln (uv) = ln u + ln v
2. loga (u / v) = loga u - loga v 2. ln (u / v) = ln u - ln v
3. loga un = n loga u 3. ln un = n ln u
2. Natural Logarithm: is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828
3. Common Logarithm: The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and also as the decimal logarithm.
4. Solving exponential and logarithmic equation:
Example 1: Solve for x : 3x = 15 . Example 2: Solve for x : log3(3x) + log3(x - 2) = 2 .
3^x=15 log3(3x) + log3(x - 2) = 2
log33x=log315 log3(3x(x - 2)) = 2
x=log315 32 = 3x(x - 2)
x=2.465 9 = 3x 2 - 6x
3x 2 - 6x - 9 = 0
3(x 2 - 2x - 3) = 0
3(x - 3)(x + 1) = 0
x = 3, - 1
2. loga (u / v) = loga u - loga v 2. ln (u / v) = ln u - ln v
3. loga un = n loga u 3. ln un = n ln u
2. Natural Logarithm: is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828
3. Common Logarithm: The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and also as the decimal logarithm.
4. Solving exponential and logarithmic equation:
Example 1: Solve for x : 3x = 15 . Example 2: Solve for x : log3(3x) + log3(x - 2) = 2 .
3^x=15 log3(3x) + log3(x - 2) = 2
log33x=log315 log3(3x(x - 2)) = 2
x=log315 32 = 3x(x - 2)
x=2.465 9 = 3x 2 - 6x
3x 2 - 6x - 9 = 0
3(x 2 - 2x - 3) = 0
3(x - 3)(x + 1) = 0
x = 3, - 1
4) Analytical Geometry
1. Ellipses: is formed by cutting a three dimensional cone with a slanted plane. center (0,0)
(x-h)^2 + (y-k)^2 =1
r^2 r^2
2. Parabolas: is formed by intersecting the plane through the cone and the top of the cone. Parabolas can be the only conic sections that are considered functions because they pass the vertical line test.
A vertical axis has a focus at (h,k+p) and the equation (x-h)2=4p(y-k). A horizontal axis has a focus at (h+p,k) and the equation (y-k)2=4p(x-h). The vertex is always halfway in between the focus and directrix at a distance p from both.
3. Hyperbola: A hyperbola is formed when a plane slices the top and bottom section of the cone. The equation for a hyperbola is
(x-h)^2 - (y-k)^2 =1
a^2 b^2
4. Need help heres a more helpful link about conics: http://www.barstow.edu/lrc/tutorserv/113ellhy.pdf
(x-h)^2 + (y-k)^2 =1
r^2 r^2
2. Parabolas: is formed by intersecting the plane through the cone and the top of the cone. Parabolas can be the only conic sections that are considered functions because they pass the vertical line test.
A vertical axis has a focus at (h,k+p) and the equation (x-h)2=4p(y-k). A horizontal axis has a focus at (h+p,k) and the equation (y-k)2=4p(x-h). The vertex is always halfway in between the focus and directrix at a distance p from both.
3. Hyperbola: A hyperbola is formed when a plane slices the top and bottom section of the cone. The equation for a hyperbola is
(x-h)^2 - (y-k)^2 =1
a^2 b^2
4. Need help heres a more helpful link about conics: http://www.barstow.edu/lrc/tutorserv/113ellhy.pdf