POLYNOMIAL, RATIONAL, EXPONENTIAL
AND
LOGARITHMIC FUNCTIONS th
4 CSE
Bilingual Program
UNIT 9: POLYNOMIAL, RATIONAL ,EXPONENTIAL AND LOGARITHMIC FUNCTIONS
1.-POLINOMIAL FUNCTIONS: Their domain is always all the real numbers. ďƒ˜ Linear Functions A function that can be graphically represented in the Cartesian coordinate plane by a straight line is called a Linear function. A very common way to express a linear function is:
(m and b are constants) The slope (also called gradient) of a straight line shows how steep a straight line is. When the gradient is 0, it is a Constant Function. Its formula is f(x) = c. Its graph is a horizontal line. To calculate the Gradient:
The gradient is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run.
(x1, y1) and (x2, y2) represent two points in the straight line. It is important to keep the x-and y-coordinates in the same order in both the numerator and the denominator otherwise you will get the wrong slope. The slope m tells us if the function is increasing, decreasing or constant:
m<0
decreasing
m=0
constant
m>0
increasing
ď&#x192;&#x2DC; Quadratic Functions A quadratic function, is a polynomial function of the form:
The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the yaxis.
Characteristics or Properties of Graphs of Quadratic Functions: 1.- The vertex of the graph of a quadratic function f(x) = ax2+ bx + c is the point
V
b b ,f 2a 2a
2.- The range of a quadratic function is: all real numbers greater than or equal to the y-value, if the vertex is a minimum. all real numbers less than or equal to the y-value, if the vertex is a maximum. 3.- They have an axis of symmetry. It is always a vertical line of the form x = n, where n is a real number 4.- To find the x-intercepts, solve the quadratic equation ax2+ bx + c = 0. To find the y-intercept of the parabola, find f(0)
2.-FUNCTIONS WHOSE GRAPH IS A HYPERBOLA ď&#x192;&#x2DC; Inversely proportional function:
The formula is
f ( x)
k x
When k = 1, it is called Reciprocal function:
f ( x)
1 x
The graph of the reciprocal function is:
Characteristics of the inversely proportional function: 1.- Its graph is a hyperbola. 2.- It is an odd function. 3.- Domain is the Real Numbers, except 0, because 1/0 is undefined:
Dom = {x
R /x
0} = R - {0}
4.- Its Range is also the Real Numbers, except 0.
Img = R - {0}
5.- The graph has a horizontal asymptote at the x –axis: y = 0 (This means the graph gets closer to the x-axis as the value of x increases, but it never meets the x- axis) 6.- The graph has a vertical asymptote at the y –axis: x = 0 (This means the graph gets closer to the y-axis as x gets closer to 0 but it never meets the y-axis)
Rational function: A rational function is defined as the quotient of two polynomial functions:
f(x)
P(x) Q(x)
If P(x) and Q(x) are first-degree polynomials, the graph is a hyperbola. For example:
y
2 x -3
4
3.- EXPONENTIAL FUNCTION Its formula is
f(x) a x , a 1 , a
0
The function value will be positive because a positive base raised to any power is positive. To graph exponential functions, remember that unless they are transformed, the graph will always pass through (0, 1) and will approach, but not touch or cross the x-axis
Characteristics of the Exponential Function: The domain is R The range is the set of strictly positive real numbers: (0, ) The function is continuous in its domain The function is increasing if a > 1 and decreasing if 0<a<1 The x-axis is a horizontal asymptote.
4.-LOGARITHMIC FUNCTION The logarithmic function is the function:
f ( x) log a x, a 1 , a 0 , and x > 0 (The function is read "log base a of x") Since x > 0, the graph of the above function will be in quadrants I and IV.
Logarithmic functions are the inverse of exponential functions. Their graphs are symmetric with respect to the angle bisector of the first quadrant.
Characteristics of the logarithmic Function: The The The The The
range is R domain is the set of strictly positive real numbers function is continuous in its domain function is increasing if a > 1 and decreasing if 0 < a < 1 negative y-axis is a vertical asymptote
Definition of logarithm:
log a x
y
ay
x
We read this in this way: "log base a of x is y”.
The two most used logarithms are called: common logarithms (base 10)
natural logarithms (base e)
PROPERTIES OF LOGARITHMS: 1) loga 1 =0 because 2) loga a =1
because
3) loga ax =x because 4) loga MN = loga M + loga N Think: Multiply two numbers with the same base, add the exponents.
5) loga M/N =logaM – logaN Think: Divide two numbers with the same base, subtract the exponents.
6) loga Mp =p·logaM