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MATH

MATH. Part 2. Linear Functions. - Graph is a line. Equation of a Line. Standard form: Ax + By = C Slope Intercept Form: y = mx + b m = slope b = y – intercept or the value of y when x is zero. Equation of a Line. Point – slope form: y – y 1 = m(x-x 1 )

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MATH

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  1. MATH Part 2

  2. Linear Functions - Graph is a line Equation of a Line • Standard form: Ax + By = C • Slope Intercept Form: y = mx + b • m = slope • b = y – intercept or the value of y when x is zero

  3. Equation of a Line • Point – slope form: y – y1 = m(x-x1) - used when a point and slope are given • Two – point form: or y – y1 = ( - given two points

  4. Slope and Orientation of Lines UNDEFINED SLOPE POSITIVE SLOPE ZERO SLOPE NEGATIVE SLOPE

  5. Parallel and Perpendicular Lines Parallel Lines Perpendicular Lines Same slope m1 = m2 m1 = -

  6. x and y intercepts y - intercept x – intercept - Value of x when y is zero - Value of y when x is zero

  7. Quadratic Equations • Equations dealing with variables whose highest exponent is 2. • Standard form: y = ax2 + bx + c

  8. Factoring • Get the Common Monomial Factor first. • After getting the CMF, use the techniques of factoring. Example: 12x4 - 48x3 - 15x2 = 3x(4x3 -16x2 – 5x)

  9. Trinomials Factoring where a = 1 x2 + bx + c = (x + m)(x + n) Wherein: m + n = b mn = c

  10. Example: x2 + 3x – 10 What are the factors of c which give a sum of b? (x – 2)(x + 5) m = -2; n = 5 m + n = 3 mn = -10

  11. Trinomials Factoring where a ≠ 1 ax2 + bx + c = (mx + n)(px +q) Wherein: mq +np = b nq = c

  12. Example: 6x2 – 5x – 6 = (3x + 2)(2x – 3) m = 3, n = 2, p = 2, q = -3 3(-3) + 2(2) = -5 2(-3) = -6

  13. Perfect Square Trinomials x2 + 2xy + y2 = (x + y)2 Example: x2 – 8x + 16 = (x – 4)2

  14. Binomials Difference of Two Squares (DOTS) x2 – y2 = (x + y)(x – y) Example: 36c2-144 = (6c + 12)(6c – 12)

  15. Sum of Two Cubes x3 + y3 = (x + y)(x2 – xy +y2) Example: y3+ 8 = (y + 2)(y2 – 2y + 4)

  16. Difference of Two Cubes x3 – y3 = (x – y)(x2 + xy + y2) Example: b3 – 64 = (b – 4)(b2 + 4b + 16)

  17. Applications of Factoring • Simplifying rational algebraic expressions or dividing polynomials • Getting the solutions/roots/zeroes of quadratic equations

  18. Quadratic Formula • An alternative way of solving for the roots/zeroes of a quadratic equation

  19. Discriminant • If b2 – 4ac < 0 - no real roots, imaginary • If b2 – 4ac = 0 - roots are real and equal • If b2 – 4ac > 0 - roots are real and unequal

  20. Laws of Exponents

  21. Exponential Functions F(x) = One to One Correspondence of Exponential Functions If xa = xb Then a = b

  22. Radicals • Exponents in fraction form * Rules of exponents also apply to radicals

  23. Rationalizing Radicals • Simplifying the radicals by “removing” the radical sign from the denominator Example: = =

  24. Adding or Subtracting Radicals • Treat radicals like variables and combine like terms/radicals Example: 6

  25. Logarithmic Functions logaN = x  N = ax Example: log232 = 5 32 = 25 • Common Logarithm • No indicated base  base is 10 • Example: • Log 10,000 = log1010000 = 4

  26. Properties of Logarithm • log xy= log x + log y • log • log xn= n log x • ln e = loge e = 1

  27. Imaginary Numbers • Let: thus:

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