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PRECALCULUS I

PRECALCULUS I. Graphs and Lines Intercepts, symmetry, circles Slope, equations, parallel, perpendicular. Dr. Claude S. Moore Danville Community College. Graph of an Equation. Equation - equality of two quantities.

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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.

  11. PRECALCULUS I • Functions and Graphs • Function, domain, independent variable • Graph, increasing/decreasing, even/odd Dr. Claude S. MooreDanville Community College

  12. Definition: Function A function f from set A to set B is a rule of correspondence that assigns to each element x in set A exactly one element y in set B. Set A is the domain (or set of inputs) of the function f, and set B contains range (or set of outputs).

  13. Characteristics of a Function 1. Each element in A (domain) must be matched with an element of B (range). 2. Each element in A is matched to not more than one element in B. 3. Some elements in B may not be matched with any element in A. 4. Two or more elements of A may be matched with the same element of B.

  14. Functional Notation Read f(x) = 3x - 4 as “f of x equals three times x subtract 4.” x inside parenthesis is theindependent variable. f outside parenthesis is the dependent variable. For the function f(x) = 3x - 4, f(5) = 3(5) - 4 = 15 - 4 = 11, and f(-2) = 3(-2) - 4 = - 6 - 4 = -10.

  15. Piece-Wise Defined Function A “piecewise function” defines the function in pieces (or parts). In the function below, if x is less than or equal to zero, f(x) = 2x - 1; otherwise, f(x) = x2 - 1.

  16. Domain of a Function Generally, the domain is implied to be the set of all real numbers that yield a real number functional value (in the range). Some restrictions to domain: 1. Denominator cannot equal zero (0). 2. Radicand must be greater than or equal to zero (0). 3. Practical problems may limit domain.

  17. Summary of Functional Notation In addition to working problems, you should know and understand the definitions of these words and phrases: dependent variable independent variable domain range function functional notation functional value implied domain

  18. Vertical Line Test for a Function A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

  19. Increasing, Decreasing, and Constant Function On the interval containing x1 < x2, 1. f(x) is increasing if f(x1) < f(x2). Graph of f(x) goes up to the right. 2. f(x) is decreasing if f(x1) > f(x2). Graph of f(x) goes down to the right. On any interval, 3. f(x) is constant if f(x1) = f(x2). Graph of f(x) is horizontal.

  20. Even and Odd Functions 1. A function given by y = f(x) is even if, for each x in the domain, f(-x) = f(x). 2. A function given by y = f(x) is odd if, for each x in the domain, f(-x) = - f(x).

  21. PRECALCULUS I • Composite and Inverse Functions • Translation, combination, composite • Inverse, vertical/horizontal line test Dr. Claude S. MooreDanville Community College

  22. Vertical Shifts(rigid transformation) For a positive real number c, vertical shifts of y = f(x) are: 1. Vertical shift c unitsupward: h(x) = y + c = f(x) + c 2. Vertical shift c units downward: h(x) = y - c = f(x) - c

  23. Horizontal Shifts(rigid transformation) For a positive real number c, horizontal shifts of y = f(x) are: 1. Horizontal shift c unitsto right: h(x) = f(x - c) ; x - c = 0, x = c 2. Vertical shift c units to left: h(x) = f(x + c) ; x + c = 0, x = -c

  24. Reflections in the Axes Reflections in the coordinate axes of the graph of y = f(x) are represented as follows. 1. Reflection in the x-axis: h(x) = -f(x)(symmetric to x-axis) 2. Reflection in the y-axis: h(x) = f(-x)(symmetric to y-axis)

  25. Arithmetic Combinations Let x be in the common domain of f and g. 1. Sum: (f + g)(x) = f(x) + g(x) 2. Difference: (f - g)(x) = f(x) - g(x) 3. Product: (f × g) = f(x)×g(x) 4. Quotient:

  26. Composite Functions The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function f with the function g is defined by (f⃘g)(x) = f(g(x)). Two step process to find y = f(g(x)): 1. Find h = g(x). 2. Find y = f(h) = f(g(x))

  27. One-to-One Function For y = f(x) to be a 1-1 function, each x corresponds to exactly one y, and each y corresponds to exactly one x. A 1-1 function f passes both the vertical and horizontal line tests.

  28. VERTICAL LINE TEST for a Function A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

  29. HORIZONTAL LINE TEST for a 1-1 Function The function y = f(x) is a one-to-one (1-1) function if no horizontal line intersects the graph of f at more than one point.

  30. Existence of an Inverse Function A function, f, has an inverse function, g, if and only if (iff) the function f is a one-to-one (1-1) function.

  31. Definition of an Inverse Function A function, f, has an inverse function, g, if and only iff(g(x)) = x and g(f(x)) = x,for every x in domain of gand in the domain of f.

  32. Relationship between Domains and Ranges of f and g If the function f has an inverse function g, then domain range f x y g x y

  33. Finding the Inverse of a Function 1. Given the function y = f(x). 2. Interchange x and y. 3. Solve the result of Step 2 for y = g(x). 4. If y = g(x) is a function, then g(x) = f-1(x).

  34. PRECALCULUS I • Mathematical Modeling • Direct, inverse, joint variations;Least squares regression Dr. Claude S. MooreDanville Community College

  35. Direct Variation Statements 1. y varies directly as x. 2. y is directly proportional to x. 3. y = mx for some nonzero constant m. NOTE: m is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find m.y = mx yields 3 = m(2) or m = 1.5. Thus, y = 1.5x.

  36. Direct Variation as nth Power 1. y varies directly as the nth power of x. 2. y is directly proportional to the nth power of x. 3. y = kxn for some nonzero constant k. NOTE: k is the constant of variation or constant of proportionality.

  37. Inverse Variation Statements 1. y varies inversely as x. 2. y is inversely proportional to x. 3. y = k / x for some nonzero constant k. NOTE: k is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find k.y = k / x yields 3 = k / 2 or k = 6. Thus, y = 6 / x.

  38. Joint Variation Statements 1. z varies jointly as x and y. 2. z is jointly proportional to x and y. 3. y = kxy for some nonzero constant k. NOTE: k is the constant of variation. Example: If z = 15 when x = 2 and y = 3,find k.y = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5. Thus, y = 2.5xy.

  39. Least Squares Regression This method is used to find the “best fit” straight line y = ax + bfor a set of points, (x,y), in the x-y coordinate plane.

  40. Least Squares Regression Line The “best fit” straight line, y = ax + b, for a set of points, (x,y), in the x-y coordinate plane.

  41. Least Squares Regression Line X Y X2 XY 1 3 1 3 2 5 4 10 4 5 16 20 å 7 13 21 33 Solving for a = 0.57 and b = 3, yields y = 0.57x + 3.

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