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Interpolation is a process used to estimate a value for a relation that lies between two known values Extrapolation is a process used to estimate a value for a relation that lies beyond the known values The equation of a line through the points (the known values) helps you interpolate and extrapolate Example: Using the data and the graph, estimate the value, in dollars, of 20 pounds Use interpolation Locate two points on either side of 20 pounds 15 pounds and 30 pounds Draw a line (or place a straight edge) through the graphed points Draw a line segment from 20 on the x-axis to that line Draw a line segment from that intersection point to the y-axis Read the scale on the y-axis 20 pounds is about 28 or 29 dollars Example: Use extrapolation to estimate the value of 50 pounds Draw a line segment from 50 on the x-axis to the line through the points Extend the line if necessary Read the corresponding value on the y-axis Extend the axis scale if necessary 50 pounds is a little more than 70 dollars

A line graph shows the linear relationship between two variables Two types of line graphs that describe motion are used frequently in physics Connecting Math to Physics The following line graph shows a changing relationship between the two graphed variables

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A linear equation can be written as a relation (or a function), y mx b, where m and b are real numbers, m represents the slope of the line, and b represents the y-intercept, the point at which the line crosses the y-axis

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The graph of a linear equation is a line The line represents all of the solutions of the linear equation To graph a linear equation, choose three values for the independent variable (Only two points are needed, but the third point serves as a check) Calculate the corresponding values for the dependent variable Plot each ordered pair (x, y) as a point Draw a best-fit line through the points Example: Graph y

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The slope of a line is the ratio of the change in y-coordinates to the change in x-coordinates It is the ratio of the vertical change (rise) to the horizontal change (run) This number tells how steep the line is It can be a positive or a negative number To find the slope of a line, select two points, (x1, y1) and (x2, y2) Calculate y the run, which is the difference (change) x between the two x-coordinates, x2 x1 y2 Calculate the rise, which is the difference y (x2, y2) (change) between the two y-coordinates, y1 (x1, y1) y Form the ratio y2 y1 Slope m where x1

If there is some nonzero constant, m, such that y mx, then y varies directly as x That means as the independent variable x increases, the dependent variable y increases The variables x and y also are said to be proportional This is a linear equation of the form y mx b in which the value of b is zero The graph passes through the origin, (0, 0) Connecting Math to Physics In the force equation for an ideal spring, F kx, where F is the spring force, k is the spring constant, and x is the spring displacement, spring force varies directly as ( is proportional to) spring displacement That is, the spring force increases as the spring displacement increases

If there is some nonzero constant, m, such that y , then y varies inversely as x x That means as the independent variable x increases, the dependent variable y decreases The variables x and y also are said to be inversely proportional This is not a linear equation because it contains the product of two variables The graph of an inverse relationship is a hyperbola This relationship can be written as y y xy Example: Graph the equation xy 90

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