Graphs and Relationships

Interpolation

-a process in which you estimate a value that lies between two known values on a graph; the process of finding a value between two points on a line or curve -look at the graph and calculate a point that lies between two known points

Uses of graphs

-analyze and understand data more completely -provide a visual depiction of the relationships -communicate results faster and clearer -decode and analyze their data and predict trends

quadratic graph

-based on a quadratic equation -parabola

inverse graph

-based on an inverse equation -when x increases, y decreases, and when x decreases, y increases

Extrapolation

-make predictions based on the visible trend to estimate a value that lies beyond the known data -most accurate for points that are close to known values and for graph segments that are close to linear -look at the plotted points and estimate a value that is beyond the graph's scope

Cartesian graph

-the grid that we commonly use to plot graphs -has an x-axis and a y-axis

Boyle's law

-the product of the pressure (P) and the volume (V) of an ideal gas at a particular temperature is always equal to a constant value, PV = constant = k -inverse graph

linear graph

-usually a diagonal straight line, but can also be a vertical or a horizontal straight line -based on a linear equation -when x increases, y also increases, and when x decreases, y decreases

parabola

direction is determined by the value of a, the coefficient of x. If a is positive, then the parabola opens upward. If a is negative, then the parabola opens downward.

α

proportional to

Interpolation Formula

start with two known points as inputs, such as (x0, y0) and (x1, y1). Then, for a point between those two known points, determine the value of one variable (y) when given the value of the other variable y-y₀ = [(y₁−y₀)/(x₁−x₀)] × (x−x₀)

x-axis

the horizontal axis on the coordinate plane

slope

the ratio of change in the y-coordinate to the change in the x-coordinate

y-axis

the vertical axis on the coordinate plane

origin

where the x-axis and the y-axis intersect, usually point (0, 0) on the Cartesian plane

linear equation

written as y = mx, where m is a constant called the slope and y varies directly with x

quadratic equation

y = ax² + bx + c, where a, b, and c are constants; when b = c = 0, the quadratic equation becomes y = ax², and the vertex (lowest point) of the parabola passes through the origin (0, 0)

inverse equation

y = k/x, where k is the nonzero constant

Extrapolation Formula

y-y₀ = [(y₁−y₀)/(x₁−x₀)] × (x−x₀)