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Section 1.2: Graphs of Equations; Circles

This section provides an in-depth exploration of graphs of equations, specifically focusing on determining if a point lies on a graph, identifying intercepts, and understanding symmetries of graphs. Helpful examples provide clarity on testing points against equations and finding x- and y-intercepts. Additionally, it covers x-axis, y-axis, and origin symmetry, explaining how to test for these properties in various equations, complete with practice problems to reinforce understanding.

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Section 1.2: Graphs of Equations; Circles

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  1. Section 1.2: Graphs of Equations; Circles

  2. Determine if the following point is on the graph of the equation. • 2x – y = 6; (2, 3) • Step 1: Plug the given points into the given equation. • 2(2) – (3) = 6 • Step 2: Simplify. • 4 – 3 = 6 • 1 = 6 • Step 3: If the answer is true, the point is on the graph. If the answer is false, then the point is not on the graph. • Since 1 =6 is false, the point (2, 3) is not on the graph of this equation.

  3. Intercepts An intercept is the point at which the graph crosses or touches one of the coordinate axes. The x-intercept is where the graph crosses the x-axis. What’s special about the coordinates of this point? • The ordinate (y value) is zero. The y-intercept is where the graph crosses the y-axis. What’s special about the coordinates of this point? • The abscissa (x value) is zero.

  4. Examples: Find the intercepts 1. x2 + y - 9 = 0 2. y =

  5. Symmetries • x – axis symmetry: for every point (x, y) on a graph, the point (x, -y) is also on the graph • y – axis symmetry: for every point (x, y) on a graph, the point (-x, y) is also on the graph • Origin symmetry: for every point (x, y) on a graph, the point (-x, -y) is also on the graph

  6. Examples: Plot each point, then plot the point that is symmetric to it with respect to the a.) x – axis, b.) y – axis, and c.) origin. 1. (-2, 1) 2. (4, -3)

  7. To Test for Symmetry • x-axis symmetry: Replace y with –yin the equation • y-axis symmetry: Replace x with –xin the equation • Origin symmetry: Replace x with–xand replace y with –yin the equation In each case, if you get the exact same equation back, you have that type of symmetry.

  8. Examples: Test for symmetry 1. x2 + y - 9 = 0 2. y = Homework: p. 18-19 #2-44 even

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