The point-slope formula
- Assume we know two things about a straight line: its slope and one of the points it goes through, where \(a\) is the slope and \(\left(x_1,\ y_1\right)\) is the point
- We can use the definition of the slope to derive the linear function for the straight line: \(a=\frac{y-y_1}{x-x_1}\)
- Multiplying both sides of this equation by \(x-x_1\), we get the point-slope formula: \(y-y_1=a\left(x-x_1\right)\)
Example
- Find the equation of the line through \(\left(-2,\ 3\right)\) with slope \(-4\).
The point-point formula
- We want to find the equation for the line passing through the points \(\left(x_1,\ y_1\right),\ \left(x_2,\ y_2\right)\). We can use the point-point formula to derive this equation.
- Write the formula for the slope: \(a=\frac{y_2-y_1}{x_2-x_1}\)
- We can now use the point-slope formula: \(y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\)
- Solving for \(y\): \(y=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)+y_1\Longrightarrow y=\frac{y_2-y_1}{x_2-x_1}x+\left(y_1-\frac{y_2-y_1}{x_2-x_1}x_1\right)\)
Example
- Find the equation for the linear function going through the points \(\left(-1,\ 3\right),\ \left(5,\ -2\right)\)
General equation for a straight line
- We start with the following formula: \(Ax+By+C=0\)
- Solving for \(y\): \(y=-\frac{A}{B}x-\frac{C}{A}\)
Linear models
- Linear models are extremely popular in mathematics as well as economics (especially in intro classes)
Example
- The European population in 1960 was 641 million. In 1970 the same figure was 705 million. Estimate a linear model of population growth in Europe.
- We have two coordinates: \(\left(0,\ 641\right),\ \left(10,\ 705\right)\)
- We can now use the point-point formula: \(P=\frac{705-641}{10}t+641=6.4t+641\)
The consumption function
- The consumption function expresses consumption in an economy as a function of the economy's output
- Mathematically, \(C=f\left(Y\right)\)
- Many models assume the consumption function to be linear: \(C=a+bY\) (this is referred to as Keynesian cross), where \(b\) is the marginal propensity to consume
Supply and demand
- The supply and demand framework is also often represented as with a linear model
- \(D:\ P=a-bQ\)
- \(S:\ P=\alpha+\beta Q\)
- We can use this linear model to identify the (partial) equilibrium price \(P^e\)
Quadratic functions
- The general form of the quadratic function is \(f\left(x\right)=ax^2+bx+c\), where \(a,\ b,\ \&\ c\).
- The graph of the quadratic function is called a parabola