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PRECALCULUS I

This resource provides a comprehensive overview of key concepts in precalculus focusing on graphs and lines. Learn how to find intercepts, test for symmetry, and apply the standard form of circles, as well as the slope-intercept and point-slope forms of linear equations. Additionally, explore the conditions for parallel and perpendicular lines, including specific equations for different line types. Perfect for students aiming to strengthen their understanding of these fundamental topics in mathematics.

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PRECALCULUS I

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  1. PRECALCULUS I • Graphs and Lines • Intercepts, symmetry, circles • Slope, equations, parallel, perpendicular Dr. Claude S. MooreDanville Community College

  2. 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

  3. Finding Interceptsof 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.

  4. 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.

  5. Standard Form of theEquation of a Circle The point (x, y) lies on the circle of radius r and center (h, k) if and only if (x - h)2 + (y - k)2 = r2 .

  6. 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).

  7. 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

  8. Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through the point (x1, y1) isy - y1 = m(x - x1).

  9. 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)

  10. Parallel and Perpendicular Lines Parallel: nonvertical l1 and l2 are parallel iff m1 = m2 and b1 ¹ b2.*Two vertical lines are parallel. Perpendicular: l1 and l2 are perpendicular iff m1 = -1/m2 or m1 m2 = -1.

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