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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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Move ahead quickly Move ahead slowly Maintain position Time
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This line graph shows a constant relationship between the two graphed variables
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Linear Equations
Math Handbook
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
Dependent variable Independent variable
Slope
y-intercept
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
1 x 3 2
Plot of Ordered Pairs
Calculate three ordered pairs to obtain points to plot
Ordered Pairs
x 0 2 6 y 3 2 0
Slope
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
Appendix A
rise run
y2 x2
y1 x1
y , x
x1 x
Math Handbook
Direct Variation
Math Handbook
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
Inverse Variation
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
m x 1 x
Ordered Pairs
x 10 6 3 2 2 3 6 10 y 9 15 30 45 45 30 15 9
8 4 40 20
Graph of an Inverse Variation
0 20
, where Connecting Math to Physics In the equation for the speed of a wave, f is wavelength, f is frequency, and v is wave speed, wavelength varies inversely as ( is inversely proportional to) frequency That is, as the frequency of a wave increases, the wavelength decreases v is constant
Math Handbook
Quadratic Graphs
Math Handbook
A quadratic relationship is a relationship of the form y ax 2 bx c, where a 0 A quadratic relationship includes the square of the independent variable, x The graph of a quadratic relationship is a parabola Whether the parabola opens upward or downward depends on whether the value of the coefficient of the squared term, a, is positive or negative Example: Graph the equation y x2 4x 1
Ordered Pairs
x 1 0 1 2 3 4 5 y 6 1 2 3 2 1 6
Graph of a Quadratic Equation
Connecting Math to Physics A position-time graph in the shape of a quadratic relation means that the object is moving at a constant acceleration
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