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Graphs of Equations

Learn to graph equations, find intercepts, and test for symmetry to solve math problems effectively. Practice key concepts step by step with examples.

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Graphs of Equations

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  1. Graphs of Equations Objectives: Graph an equation. Test for symmetry.

  2. Equation in two variables • Equation in two variables - Statement in which two expressions involving x and y are equal. • The statement may be true or false based on the numbers put in for the variables. • Any values that make the statement true are said to “satisfy” the equation. • There are many pairs of values that “satisfy” the equation. The best way to show all possible solutions is graphically.

  3. Graph of an equation consists of the set of points that satisfy the equation. Graphing by PLOTTING POINTS • Find the intercepts X intercept: Substitute a zero in for y and solve for x. ALSO called ROOTS and ZEROS Y intercept: Substitute a zero in for x and solve for y. • Select other values of x around intercepts and solve for the y’s • Graph all points • Connect

  4. Graph: x = y2 – 2y - 3 • X intercept (Zero): x = 02 – 2(0) – 3 = -3 (-3, 0) • Y intercepts: 0 = y2 – 2y – 3 0 = (y – 3)(y + 1) 0 = y – 3 0 = y + 1 y = 3 y = -1 (0, 3), (0, -1)

  5. Graph: x = y2 – 2y - 3 • (-3,0) • (0,3) (0, -1) • Now that the intercepts have been found select several x’s or y’s and put them into the equation and find the x or y. • ( , -2 ), ( ,-3), ( , 1 ), ( , 2 ), ( , 4 ), ( , 5) • Put each value into the equation and find the x. • (5, -2), (12, -3), (-4, 1), (-3, 2), (5,4) • Graph points and connect

  6. Using Technology to graph • Solve the equation for y. Remember if you take the square root you will have two equations – one positive and one negative. • Put equation/s into the calculator in graph mode. • Use trace option to find points on the curve. • Transfer exact points to hard copy.

  7. Graph: x = y2 – 2y - 3 • Solve for y Complete the square to get y by itself. • X + 3 = y2 – 2Y Move constant • X + 3 + 1= Y2 – 2Y + 1 Half the linear term2 • X + 3 = (Y – 1)2 Factor • SQRT(X + 3)= Y – 1 Take square root • -SQRT (X + 3) = Y – 1 • SQRT(X + 3) + 1 = Y Add 1 • -SQRT (X + 3) + 1 = Y

  8. SQRT(X + 3) + 1 = Y • -SQRT (X + 3) + 1 = Y • Go to graph mode on the calculator. • Type in the two equations formed when y was by itself. • Check the range. Start with standard range and see if the graph is complete. • Use trace mode to find exact graph values.

  9. Tests for symmetry • X-axis: Replace each y in the equation with a –y. If the result is the original equation the equation is symmetric to the x-axis • Y-axis: Replace each x in the equation with a –x. If the result is the original equation then it is symmetric to the y-axis • Origin: Replace each x with a –x and each y with a –y, if the original equation results it is symmetry to the origin.

  10. Y = 2/x • Intercepts: X- intercept: 0 = 2/x , can never equal zero, so none Y-intercept: Y = 2/0: cannot occur so none • Other points: (-1, ), (-2, ), (-3, ), (1, ), (2, ), (3, ) (-1, -2), (-2, -1), (-3, -2/3), (1, 2), (2, 1), (3, 2/3) • Graph and connect

  11. Test y=2/x for symmetry • X-axis: Put –y in for y -y = 2/x => y = -2/x Not the same as original so not symmetrical to x-axis. • Y-axis: Put –x in for x Y = 2/-x => y = -2/x Not the same as original so not symmetrical to y-axis • Origin: Put –y in for y and –x in for x -y = 2/-x => y = 2/x Same as original so is symmetrical to origin.

  12. Assignment • Page 174 • #11, 15, 19, 29, 33, 39, 43

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