Chapter 8: Graphs and Functions

Rectangular Coordinate System 8.1

Rectangular Coordinate System 8.1 Rectangular Coordinate System

Distance and Midpoint Formulas 8.1  

Circles 8.1  

Lines and Slopes 8.2 Equations of the form Ax + By = C can be visualized as a straight line Slope is rise/run x-intercept: set y = 0 y-intercept: set x = 0

Equations of Straight Lines 8.2 & 8.3 Given the slope m and the y-intercept b, the slope-intercept form is y = mx + b Given a point (x1,y1) and the slope m, the point-slope form is y-y1 = m(x-x1)

Parallel and Perpendicular 8.2 Parallel lines have the same slope Ex: y = 2x + 1 and y = 2x – 4 Perpendicular lines have slopes that are negative reciprocals Ex: y = 2x + 1 and y = -(1/2)x +3

Functions A relation is a set of ordered pairs 8.4 A relation is a set of ordered pairs A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component Graph of a function obeys the vertical line test: any vertical line crosses at most once

Domain and Range 8.4 When ordered pairs are of the form (x,y), x is the independent variable and y is the dependent variable The domain is the set of all values of the independent variable x The range is the set of all values of the dependent variable y

Linear Functions A function that can be written in the form 8.4 A function that can be written in the form f(x) = mx + b for real numbers m and b is a linear function. Example: cost and revenue models

Quadratic functions A function f is a quadratic function if 8.5 A function f is a quadratic function if f(x) = ax2 + bx + c where a, b, and c are real numbers with a not equal to 0.

Graphing Quadratic Functions 8.5 The graph of the quadratic function defined by f(x) = a(x-h)2 + k, a not 0, is a parabola with vertex (h,k) and the vertical line x = h as axis of symmetry The graph opens up if a is positive and down if a is negative The graph is wide if |a|<1 and narrow if |a|>1 compared to y = x2

More Graphing Quadratics 8.5 f(x) = ax2 + bx + c Decide if graph opens up or down Find y-intercept by setting x = 0 Find x-intercept by solving f(x) = 0 Find vertex: x = -b/(2a) Complete the graph

8.5 #41 Steve has 100 meters of fencing material to enclose a rectangular exercise run for his dog. What width will give the enclosure the maximum area?

8.5 #47