Week 1
Linear functions: y=a+b∗x
x = independent variable
y = dependent variable
b = intercept
y 2− y 1
a = slope = with two points P ( x 1 ; y 1 )∧Q( x2 ; y 2)
x 2−x 1
−b
Intersection point with horizontal axis: x=
a
Domain: Set of values x can obtain
Range: Set of values y can obtain
Important: The domain and the range of a function are determined both by
mathematical conditions and by economic meaningfulness!
2
Quadratic functions: f ( x )=a x +bx +c
If a > 0 Convex (valley-shaped)
If a < 0 Concave (hill-shaped)
( )
2
−b −b b
Extreme Values: x Extrema = ∧f =c−
2a 2a 4a
Roots: Zeros (f(x) = 0) , Intersections points with the x-axis
√ 2
Abc Formula: x 1,2=−b − +¿ ¿ ¿ b −4 ac ¿
2a
Discriminant (D) = b 2−4 a
D > 0 two distinct roots
D = 0 one root
D < 0 no roots
To find the common points of function f(x) and g(x) just set them equal (f(x) = g(x)).
Solving systems of linear equations:
1. Write the system in standard form (Put both variables on one side)
2. Solve by eliminating one of the variables of one equation
3. Solve equation left with only one variable
, Week 2
−1
Inverse function f ( y ): Describe x as a function of y
The inverse from f(x) realizes that y is the independent an x the dependent variable
We switch y-axis and x-axis
Mirroring around the line y = x
x
Exponential functions: f ( x )=c∗a with a > 0 and c as a constant
Logarithmic function: f ( x )=c∗log ( x ) with c as a constant
Important: Taking logarithms is the inverse operation of taking exponents
(Hint: In QM1 the log(x) is rarely used. If you have the choice use the ln(x), which is the
logarithm to the base of e)
Algebraic manipulations for exponents
1. a 0=1
2. a x+ y =a x∗a y
−x1
3. a = x
=¿
a
x
x− y a
4. a = y
a
Algebraic manipulations for logarithms
1. log a ¿
2. log a ¿
1
3. log a ( )=−log a x
x
4. log a
x
y ()
=log a (x)−log a ( y )
5. log a (x¿ ¿ b)=b∗log a ( x )¿
(Same rules apply for the use of ln(x); only e is the base instead of a)
The number e: e=lim
n→∞
¿¿
New functions from old
Translation: Shift of a graph
Rescaling: Multiplying the function or the argument by a constant
g ( x )=f ( x ) +a
Vertical translation, up- or (downward) with a > 0 (a < 0)
Locations of zeros will change
Linear functions: y=a+b∗x
x = independent variable
y = dependent variable
b = intercept
y 2− y 1
a = slope = with two points P ( x 1 ; y 1 )∧Q( x2 ; y 2)
x 2−x 1
−b
Intersection point with horizontal axis: x=
a
Domain: Set of values x can obtain
Range: Set of values y can obtain
Important: The domain and the range of a function are determined both by
mathematical conditions and by economic meaningfulness!
2
Quadratic functions: f ( x )=a x +bx +c
If a > 0 Convex (valley-shaped)
If a < 0 Concave (hill-shaped)
( )
2
−b −b b
Extreme Values: x Extrema = ∧f =c−
2a 2a 4a
Roots: Zeros (f(x) = 0) , Intersections points with the x-axis
√ 2
Abc Formula: x 1,2=−b − +¿ ¿ ¿ b −4 ac ¿
2a
Discriminant (D) = b 2−4 a
D > 0 two distinct roots
D = 0 one root
D < 0 no roots
To find the common points of function f(x) and g(x) just set them equal (f(x) = g(x)).
Solving systems of linear equations:
1. Write the system in standard form (Put both variables on one side)
2. Solve by eliminating one of the variables of one equation
3. Solve equation left with only one variable
, Week 2
−1
Inverse function f ( y ): Describe x as a function of y
The inverse from f(x) realizes that y is the independent an x the dependent variable
We switch y-axis and x-axis
Mirroring around the line y = x
x
Exponential functions: f ( x )=c∗a with a > 0 and c as a constant
Logarithmic function: f ( x )=c∗log ( x ) with c as a constant
Important: Taking logarithms is the inverse operation of taking exponents
(Hint: In QM1 the log(x) is rarely used. If you have the choice use the ln(x), which is the
logarithm to the base of e)
Algebraic manipulations for exponents
1. a 0=1
2. a x+ y =a x∗a y
−x1
3. a = x
=¿
a
x
x− y a
4. a = y
a
Algebraic manipulations for logarithms
1. log a ¿
2. log a ¿
1
3. log a ( )=−log a x
x
4. log a
x
y ()
=log a (x)−log a ( y )
5. log a (x¿ ¿ b)=b∗log a ( x )¿
(Same rules apply for the use of ln(x); only e is the base instead of a)
The number e: e=lim
n→∞
¿¿
New functions from old
Translation: Shift of a graph
Rescaling: Multiplying the function or the argument by a constant
g ( x )=f ( x ) +a
Vertical translation, up- or (downward) with a > 0 (a < 0)
Locations of zeros will change
