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PRECALCULUS I Graphs and Lines •Intercepts, symmetry, circles •Slope, equations, parallel, perpendicular Dr. Claude S. Moore Danville Community College 1 Graph of an Equation Equation - equality of two quantities. Solution - (a,b) makes true statement when a and b are substituted into equation. Point-plotting method - simplest way to graph. x -2 -1 0 1 2 y = 2x - 3 -7 -5 -3 -1 1 Finding Intercepts of an Equation The x-intercept is point where graph touches (or crosses) the x-axis. The y-intercept is point where graph touches (or crosses) the y-axis. 1. To find x-intercepts, let y be zero and solve the equation for x. 2. To find y-intercepts, let x be zero and solve the equation for y. Tests for Symmetry 1. The graph of an equation is symmetric with respect to the y-axis if replacing x with -x yields an equivalent equation. 2. The graph of an equation is symmetric with respect to the x-axis if replacing y with -y yields an equivalent equation. 3. The graph of an equation is symmetric with respect to the origin if replacing x with -x and y with -y yields an equivalent equation. Standard Form of the Equation of a Circle The point (x, y) lies on the circle of radius r and center (h, k) if and only if (x - 2 h) + (y - 2 k) = 2 r . Slope-Intercept Form of the Equation of a Line The graph of the equation y = mx + b is a line whose slope is m and whose y-intercept is (0, b). Definition: Slope of a Line The slope m of the nonvertical line through (x1, y1) and (x2, y2) where x1 is not equal to x2 is y2 y1 m x2 x1 Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through the point (x1, y1) is y - y1 = m(x - x1). Equations of Lines 1. General form: 2. Vertical line: 3. Horizontal line: 4. Slope-intercept: 5. Point-slope: 1. Ax + By + C = 0 2. x = a 3. y = b 4. y = mx + b 5. y y1 = m(x x1) Parallel and Perpendicular Lines Parallel: nonvertical l1 and l2 are parallel iff m1 = m2 and b1 b2. *Two vertical lines are parallel. Perpendicular: l1 and l2 are perpendicular iff m1 = -1/m2 or m1 m2 = -1.