Linear Functions with Negative Directional Coefficients

Linear Function with Positive Directional Coefficient: Analysis of Directional Change Name: Alvinska Hebat Grade: 910-Math Definition of Linear Function • • Linear function is also called a straight line function, which is a form of function that has a characteristic where the independent variable is a power of one. It is said to be a straight line function because if the position of the points is described, it will form a straight line General Form of Linear Functions with One Free Variable • • • • • The general form of a linear function equation with one independent variable is as follows: Y = a + mX or Y = mX + a where: Y = dependent variable X = independent variable a,m = constant/coefficient; m ≠ 0 a = intercept; a ≥ 0 or a ≤ 0 m = coefficient of the direction of the line / slope function / slope / gradient, m = ∆Y / ∆X From the equation above, the intercept or "a" is the value of Y when the value of X = 0, while the coefficient of the line direction or "m" is the ratio between changes in Y to change in X or (ΔY/ΔX). Graphically (in the figure) "m" is the magnitude of the slope / direction coefficient / slope of the function line Y = a + mX The linear function with a positive coefficient direction can be described as follows The figure above has a positive direction coefficient (m1 > 0) or α = (ΔY/ΔX) > 0. This means that the change in Y with the change in X is in the same direction, meaning that if X goes up then Y will go up, and if X goes down then Y will go down. The linear function with a negative coefficient direction can be described as follows: • The figure above has a negative direction coefficient (m1 < 0) or β = (ΔY/ΔX) < 0. This means that the change in Y with the change in X is in the opposite direction, meaning that if X increases, Y will decrease, and if X decreases, Y will increase. Drawing Linear Functions A linear equation can be drawn on a Cartesian plane by specifying just two coordinates. For example, describe the function Y = f(X) = 2 + 3X by trial and error: If X = 0 then Y = 2 + 3(0) = ...................... 2coordinates of point A (0,2) If X = 5 then Y = 2 + 3(5) = .......................... 17coordinates of point B (5,17) Note: the value of X is assumed to be free. The function Y = 2 + 3X is drawn by joining the coordinates of point A and the coordinates of point B that satisfy the equation ▪ In drawing curves on the Cartesian plane, it is necessary to pay attention to the element of proportionality regarding the magnitude of the variables interacting in the function. ▪ We don't necessarily have to draw the curve in the proportion of the magnitude of the variables in the proportion X : Y = 1 : 1, but we should pay attention to the magnitude of each variable. For example, if X is in millions of units while Y is in tens of units, we do not need to draw the relationship between the two variables on a scale (proportion) of 1: 1. Determining Linear Equations The equation of the linear function Y = f(X) will be known conditionally: ✓ if two coordinates are known or ✓ when one coordinate and the direction coefficient of the line are known. Determining the Linear Function Equation When Two Coordinates Are Known The linear function equation Y = f(X) = a + bX can be known when two coordinates are known. The two coordinates are for example point A (X1 , Y1 ) with point B (X2 , Y2 ) With these two coordinates of points A and B, the linear equation can be found with the formula : Determining the Equation of a Linear Function Given One Coordinate and the Coefficient of the Direction of a Straight Line If one point or coordinate (X1, Y1 ) and the coefficient of the direction of the straight line (m) are known, the equation of the linear function Y = f(X) can be found through : Y - Y1 = m(X X1 ) or Y = Y1 + m(X - X1 ) where m = ∆Y/∆X Examples of Linear Function Usage in Economic Concepts 1) Linear Trend Function 2) Consumption and savings function 3) Receiving function 4) Profit function 5) Budget Line Function 6) Demand function 7) Supply function Linear Trend Function A linear trend is a linear function where the independent variable is time while the dependent variable is a certain quantity. Trend functions are used to predict the magnitude of a certain variable at a certain time. The Linear Trend function can be written: Y = a + mt where Y is the dependent variable and t is time. From the trend equation, "a" is the price of Y at time t = 0, while "m" is the amount of change in Y over time. Consumption and Savings Function If we suppose that the amount of consumption is only affected by the level of income, then the linear consumption function is: C = a + bY where: C = consumption value (in money value); Y = income value (in money value), a = intercept / constant or autonomous consumption b = coefficient of direction / slope of the consumption function = ΔC / ΔY or MPC (Marginal Propensity to Consume) or marginal desire to consume. If savings (S) is the difference between income (Y) and consumption (C) or S = Y - C, then the savings function equation is : S = -a + (1-b)Y where: S = savings and (1-b) = coefficient of direction / slope of the savings function = ΔS / ΔY or MPS (Marginal Propensity to Save) or marginal saving intention