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Study Pages. MAT170 SPRING 2009 Material for 1 st Quiz. How to “complete the Square” for f (x) = 3x 2 + 27x - 24 :. 1. Move anything w/o an x to the left f(x) + 24 = 3x 2 + 27x 2. Make the number in front of x 2 equal to 1 (f(x) + 24) /3 = x 2 + 9x

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Study Pages

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  1. Study Pages MAT170 SPRING 2009 Material for 1st Quiz

  2. How to “complete the Square”for f(x) = 3x2 + 27x - 24: 1. Move anything w/o an x to the leftf(x) + 24 = 3x2 + 27x 2. Make the number in front of x2 equal to 1(f(x) + 24)/3 = x2 + 9x 3.Take the number in front of x, divide by 2 and then square the result. (9/2)2 4. Add the result of step 3 to both sides. (f(x) + 24)/3 + (9/2) 2 = x2 + 9x + (9/2)2 5.Factor the right side. (f(x) + 24)/3 + 20.25 = [x - (9/2)]2 6. Move everything back to the right and now you have the standard quadratic function equation. f(x) = 3[x - (9/2)]2 – 84.75

  3. What is meant by an ODD function? What is meant by an EVENfunction? f(x) = f(-x) -f(x) = f(-x)

  4. What is the Standard form of aQuadratic Function? f(x) = a(x-h)2+k am0 The graph of f is a parabola with vertex at point (h,k) if a>0, it opens UPward; if a<0, it opens DOWNward

  5. In a Standard Quadratic Function, What are the formulas for the points on the Vertex (h, k) ? (h, k) =(-b , c-b2) 2a 4a

  6. What is the quadratic Formula? X= -b kpb2-4ac 2a For standard quadratic equations: ax2 + bx + c = 0 am0

  7. 4 Steps For Finding Inverse Function: • 1. Change the function notation : f(x)  y • 2. Change all the xs to ys and ys to xs • 3. Solve for Y • 4. Replace y with f-1(x)

  8. Domain Questions • Does the function have a denominator? • Does the function have a square or even root? • Does the function have a log or ln in it? • Did the function arise from finding an inverse? • Is this a “real world” problem?

  9. Shifts(addition or subtraction to the function) • Horizontal shifts (inside parentheses) • f(x + c) means the vertex moves left ( ) c units • f(x - c) • means the vertex moves right ( ) c units • Vertical Shifts (outside parentheses) • f(x)+ cmeans the vertex moves up c • f(x) - c • means the vertex moves down c

  10. Reflection of a Functionaround the X axis -f(x) (outside parentheses) Will reflect around the X axis

  11. Reflection of a Functionaround the Y axis f(-x) (inside parentheses) Will reflect around the Y axis

  12. Stretch/Shrink(multiplication or division of the function) • Vertical(outside parentheses) c f(x) • When c > 1 then stretch • When 0< c < 1 then shrink • Horizontal(inside parentheses) f(cx) • When c > 1 then shrink • When 0< c < 1 then stretch

  13. We use COMPLEX NUMBERS to remove negative numbers from inside radicals. p-1 = i and i2 = -1

  14. How do you figure the exponents of i ? Divide the exponent by 4 If it divides evenly by 4, then the answer is 1. The repeating pattern is: -1, -i, 1, i i2 =-1i3=-ii4=1i5=i

  15. What is a complex conjugate? Fora+bi, it isa-bi

  16. How do you divide by a complex number? Multiply both numerator and denominator by the complex conjugateof the denominator. 3-2i(3-2i)(4+5i) 4-5i = (4-5i)(4+5i)

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