MATHS

• what is the quadratic formula


  this is the formula

  
• what is the formula to find the length when given two points


  \sqrt{\left(x2-x1\right)^2+\left(y2-y1\right)^2}
• -xrl and -fupd meaning

  >f(-x) is reflection in the y -axis and it flips the graph from right to left

  >-f(x) is reflection in the x-axis and it flips the graph upside down.
• ey^x and yey^x(ey and yey)

  when differentiating e^x you multiply the constant of the exponent by the e
• what are the log rules


  these are the log rules

  
• what is the concept of natural logs

  >y=e^x is the inverse of lnx
• give a simple example of natural logarithms

  >basically if you see an equation involving e^x , ln(natural log) boths sides, the ln cancels out the e, to leave x remaining and vice versa
• more on natural logarithms


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• more on natural logarithms


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• even more on natural logarithms


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• what does lne equal


  >1
• what is the standard formulae for an exponential model

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  A) starting number

  B) a constant of t , which we need to find

  
• what is the general rule for logarithm

  >a^n=b

  >loga^b=n
• what are the conditions for discriminants?

  >If b2 – 4ac > 0 then the quadratic function has two distinct real roots. - If b2 – 4ac = 0 then the quadratic function has one repeated real root. - If b2 – 4ac < 0 then the quadratic function has no real roots.
• what does it mean when a curve has no stationary point?

  > it means that there is no point where the derivative is equal to zero
• what do you do then if there is no stationary point?

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  • In the context of curves, knowing the number of roots tells you how many stationary points the curve has. 
  • For example, if you're analyzing a quadratic function and want to know if it has any turning points (stationary points that are minimums or maximums), you can simply calculate the discriminant.
• what are roots in quadratics?

  >In a quadratic equation, roots are the values of 'x' that make the equation true, also known as solutions or zeros. Geometrically, these roots represent the x-intercepts of the graph of the quadratic function, where it crosses the x-axis. 
• solve this


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• What is the goal of part d in the problem?

  To find x-intercepts and y-intercept
• What is the function given in the problem?

  f(x) = -2x^3 + 15x^2 - 39x + 44
• How do you express f(0.2x) using the original function?

  f(0.2x) = -2(0.2x)^3 + 15(0.2x)^2 - 39(0.2x) + 44
• What is the factored form of f(0.2x)?

  f(0.2x) = (0.2x - 4)(-2(0.2x)^2 + 7(0.2x) - 11)
• How do you find the y-intercept of the function?

  Set x = 0 in f(0.2x)
• What is the y-intercept value when x = 0?

  y = 44
• What is the coordinate of the y-intercept?

  (0, 44)
• How do you find the x-intercepts of the function?

  Set f(0.2x) = 0
• What is the first equation to solve for x-intercepts?

  0.2x - 4 = 0
• What is the solution for x from the equation 0.2x - 4 = 0?

  x = 20
• What is the second equation to solve for x-intercepts?

  -2(0.2x)^2 + 7(0.2x) - 11 = 0
• What substitution is made to simplify the equation -2(0.2x)^2 + 7(0.2x) - 11 = 0?


  Let u = 0.2x
• What is the quadratic equation after substitution?

  -2u^2 + 7u - 11 = 0
• How do you solve the quadratic equation -2u^2 + 7u - 11 = 0?

  Using the quadratic formula
• What is the quadratic formula used to solve for u?

  u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
• What are the values of a, b, and c in the equation -2u^2 + 7u - 11 = 0?

  a = -2, b = 7, c = -11
• What is the discriminant of the quadratic equation -2u^2 + 7u - 11 = 0?

  49 - 88 = -39
• What does a negative discriminant indicate about the roots?

  No real roots exist
• What is the only real x-intercept found in the problem?

  (20, 0)
• What are the steps to find the x-intercepts and y-intercept of the function f(0.2x)?

  1. Set f(0.2x) = 0 for x-intercepts.
  2. Solve for x using the factored form.
  3. Set x = 0 to find the y-intercept.
  4. Calculate f(0) to find y-intercept value.
• What are the key points derived from the function f(0.2x)?

  • Y-intercept: (0, 44)
  • X-intercept: (20, 0)
  • No real roots from the quadratic equation