Chapter 8: Graphs and Functions

Rectangular Coordinate System 8.1

Rectangular Coordinate System 8.1

Distance and Midpoint Formulas 8.1

Circles 8.1

Lines and Slopes 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 8.2

Equations of Straight Lines Given the slope m and the y-intercept b, the slope-intercept form is y = mx + b Given a point (x 1,y 1 ) and the slope m, the point-slope form is y-y 1 = m(x-x 1 ) 8.2 & 8.3

Parallel and Perpendicular 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

Functions 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 8.4

Domain and Range 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 8.4

Linear Functions 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 8.4

Quadratic functions A function f is a quadratic function if f(x) = ax 2 + bx + c where a, b, and c are real numbers with a not equal to

Graphing Quadratic Functions 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 compared to y = x 2 8.5

More Graphing Quadratics f(x) = ax 2 + bx + c 1.Decide if graph opens up or down 2.Find y-intercept by setting x = 0 3.Find x-intercept by solving f(x) = 0 4.Find vertex: x = -b/(2a) 5.Complete the graph 8.5

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

8.6 Exponential Functions

Goes through (0,1) If b >1, then goes up from left to right If 0<b<1, then goes down from left to right x-axis is horizontal asymptote Domain is all numbers Range is y > 0

Compound Interest

Natural Exponent e

Logarithmic Functions

Exponential and Logarithmic Functions are Inverses

Useful properties