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S TUDY P AGES MAT170 SPRING 2009 Material for 1 st Quiz

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H OW TO “ COMPLETE THE S QUARE ” 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 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 = x 2 + 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

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W HAT IS MEANT BY AN ODD FUNCTION ? EVEN W HAT IS MEANT BY AN EVEN FUNCTION ?

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W HAT IS THE S TANDARD FORM OF A Q UADRATIC F UNCTION ? 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

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I N A S TANDARD Q UADRATIC F UNCTION, W HAT ARE THE FORMULAS FOR THE POINTS ON THE V ERTEX ( H, K ) ?

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W HAT IS THE QUADRATIC F ORMULA ? For standard quadratic equations: ax 2 + bx + c = 0 a 0

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4 S TEPS F OR F INDING I NVERSE F UNCTION : f(x) y 1. Change the function notation : f(x) y 2. Change all the x s to y s and y s to x s 3. Solve for Y f -1 (x) 4. Replace y with f -1 (x)

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D OMAIN Q UESTIONS 1. Does the function have a denominator? 2. Does the function have a square or even root? 3. Does the function have a log or ln in it? 4. Did the function arise from finding an inverse? 5. Is this a “real world” problem?

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Vertical Shifts (outside parentheses) f(x) + c means the vertex moves up c f(x) - c means the vertex moves down c Horizontal shifts (inside parentheses) f(x + c) means the vertex moves left ( ) c units f(x - c) means the vertex moves right ( ) c units

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(outside parentheses) Will reflect around the X axis

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(inside parentheses) Will reflect around the Y axis

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Vertical (outside parentheses) c f(x) When c > 1 then stretch When 0< c < 1 then shrink Horizontal (inside parentheses) f(cx)f(cx) When c > 1 then shrink When 0< c < 1 then stretch

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TRUE or FALSE? COMPLEX NUMBERS W E USE COMPLEX NUMBERS TO REMOVE NEGATIVE NUMBERS FROM INSIDE RADICALS. -1 = i and i 2 = -1

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H OW DO YOU FIGURE THE EXPONENTS OF i ? If it divides evenly by 4, then the answer is 1. The repeating pattern is: - 1, -i, 1, i i 2 =-1 i 3 =-i i 4 =1 i 5 =i

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W HAT IS A COMPLEX CONJUGATE ?

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H OW DO YOU DIVIDE BY A COMPLEX NUMBER ? 3-2i (3-2i)(4+5i) 4-5i = (4-5i)(4+5i)

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ax² + bx + c = 0 x² + 8x + 16 = f(x) To make the chart, you simply take any number and plug it in for x in the equation and the value you get is the y.

ax² + bx + c = 0 x² + 8x + 16 = f(x) To make the chart, you simply take any number and plug it in for x in the equation and the value you get is the y.

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