Concept Map

Concept Map

1. Chapter 1

   • Linear Models- A linear model is an equation of the form y=A+Bx. In this model, A is the initial value of y, That is the value of y when x is 0 and B is the constant change in y (Increase or Decrease) For each increase in x. Ex: y=7+3x
   • First Differences- The first differences are the differences in successive outputs and if the first difference is constant, then there is a linear model for the data.
   • Dependent and Independent variables- A variable y is a function of x if each input has exactly one output. If y is a function of x, then x is called the independent variable and y is called the depedent variable Ex: x;(1,2,3) y:(3,2,1)
   • Vertical Line Test- A graph is the graph of a function if and only if no vertical line intersects the graph more than once. Ex; Pass (1,3), (3,4), Fail(3,2),(3,4)
   • Function Notation- A function f is a rule that assigns to each input exactly one output. If we write x for the input and y for the output, then we use the following notation to describe f. f(x)=y Ex: f(3)=y
   • Net Change- The net change in the value of a function f as x changes from a to b (where a is less than or equal to b) is given by f(b)-f(a)
   • Domain and Range (graph)- The domain and range of a function f are represented in the graph as Range (y-value), Domain(x-value). Ex; (3,2)
   • Local Maximum and minimum values- The function f(a) is a local maximum value of f if, f(a)> or = f(x) for values of x near a. The function value f(a) is a local minimum value of f if, f(a)< or =f(x) for values of x near a.

2. Chapter 2

   • Increasing and Decreasing Functions- The function f is increasing if the value of f(x) increase as x increases, f is increasing on an interval I If f(a) < f(b) whenever a < b in I, and decreasing if f(a) > f(b) whenever a < b in I.
   • Average rate of change in a function- The average rate of change of the function y=f(x) between x = a and x = b is, Average rate of change= Net change in y/ change in x = f(b) - f(a)/ b - a
   • Average Speed of a moving Object- For a moving object, let s(t) be the distance it has traveled at time t. Then the average rate of change of the function s from t1 to t2 is called the average speed: Average speed= net change in distance/ change in time = s(t2) - s(t1),t2 -t2.
   • Linear Functions- A linear function is a function of the form f(x) = b + mx, where b and m are real numbers. The graph of a linear function is a line. Ex: f(x) = 3 + 4x
   • Linear functions and rate of change- Let f the linear function f(x) = b + mx. The rate of change of f is the constant m. The initial value of f is the constant b. Ex: f(x) = 2+ 5x
   • Linear Functions and slope- Let f be the linear function f(x) = b + mx. The graph of f is line with slope m. The y intercept of the graph f is the constant b. f(x) = 2 + 6x
   • Slope of a line- If (x1 y1) and (x2 y2) are different points on a line graphed in a coordinate plane, then the slope of the line is defined by; Slope = rise/run = change in y/ change in x = y2-y1/ x2-x1, Ex: 10-4/6-3 slope= 2
   • Slope and rate of change- For the linear function f(x) = b + mx, slope = rate of change = m. Ex: f(x) b + 3x slope =3
   • Slope intercept form of the equation of a line- An equation of the line that has slope m and y-intercept b is, y= b + mx
   • Point slope form of the equation of a line- An equation of the line that passes through the point (x1 y1) and has a slope m is y-y1=m(x-x1)
   • Horizontal and Vertical lines- An equation of the vertical line through (a,b) is x = a. An equation of the horizontal line through (a,b) is y =b. Ex: y=6 ; x=9
   • General form of the equation of a line- The graph of every general linear equation Ax + By + C = 0, is a line (where not both A and B are 0). Conversely, every line has an equation of this form. Ex: 2x - 3y - 12 =0
   • Parallel Lines- Suppose that two nonvertical lines have slopes m1 and m2. The lines are parallel if and only if their slopes are the same: m1=m2; Ex:2 = 2
   • Perpendicular lines- Suppose that two nonvertical lines have slopes m1 and m2. The lines are perpendicular if and only if their slopes are negative reciprocals: m1= -1/m2; Ex: 2 = -1/2
   • Direct Proportionality- We say that the variable y is directly proportional to the variable x (or y varies directly as x) if x and y are related by and equation of the form, y=kx, The constant k is called the constant of proportionality. Ex: y=12x
   • Intersection Points of Linear Functions- To find where the graph of the linear functions y= b1 + m1x and y=b2 + m2x, intersect, we solve for x in the equation b1 + m1x = b2 + m2x.
   • Number of solutions of a linear system in two variables- For a system of linear equations in two variable, exactlky one of the following is true. 1) The system has exacly 1 solution. 2) The system has no solution. 3) The system has infinitely many solutions.

3. Chapter 7.1

   • Solving systems of linear equations in two variables- A solution to a system of equations in two variables is an ordered pair of numbers that satisfies both, y = x + 1, y = x + 5