Complete the Square to Find Extrema - Algebra 1

1. What is the axis of symmetry formula for $f(x)=ax^2+bx+c$ in standard form?

   $x=-\frac{b}{2a}$. This formula finds the vertex $x$-coordinate directly.

2. What condition on $a$ makes the vertex of $f(x)=a(x-h)^2+k$ a maximum?

   $a<0$. Negative $a$ makes the parabola open downward.

3. What is the maximum value of $f(x)=-x^2+2x+8$?

   $9$. The $k$ value in vertex form gives the extremum.

4. What is the minimum value of $f(x)=x^2+4x+10$?

   $6$. The $k$ value in vertex form gives the extremum.

5. What is the minimum value of $f(x)=a(x-h)^2+k$ when $a>0$?

   $k$. The $y$-coordinate of the vertex is the minimum.

6. What is the vertex form of $f(x)=4x^2-4x+1$ after completing the square?

   $f(x)=4\left(x-\frac{1}{2}\right)^2$. Factor out $4$: $4\left[\left(x-\frac{1}{2}\right)^2-\frac{1}{4}\right]+1=4\left(x-\frac{1}{2}\right)^2$.

7. What number is added and subtracted to complete the square for $x^2+bx$?

   $\left(\frac{b}{2}\right)^2$. Take half the coefficient of $x$ and square it.

8. What is the minimum value of $f(x)=x^2+10x+16$?

   $-9$. The $k$ value in vertex form gives the extremum.

9. What is the vertex form of $f(x)=-3x^2-12x+6$ after completing the square?

   $f(x)=-3(x+2)^2+18$. Factor out $-3$: $-3[(x+2)^2-4]+6=-3(x+2)^2+18$.

10. What is the minimum value of $f(x)=x^2-2x-3$?

    $-4$. The $k$ value in vertex form gives the extremum.

11. What is the minimum value of $f(x)=3x^2-12x+5$?

    $-7$. The $k$ value in vertex form gives the extremum.

12. What is the vertex form of $f(x)=x^2+4x+10$ after completing the square?

    $f(x)=(x+2)^2+6$. Complete the square: $(x+2)^2-4+10=(x+2)^2+6$.

13. What is the vertex form of $f(x)=x^2-12x+20$ after completing the square?

    $f(x)=(x-6)^2-16$. Complete the square: $(x-6)^2-36+20=(x-6)^2-16$.

14. What is the completed-square form of $x^2+bx$ (no constant term) ?

    $\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4}$. Add and subtract $\left(\frac{b}{2}\right)^2$ to complete the square.

15. What condition on $a$ makes the vertex of $f(x)=a(x-h)^2+k$ a minimum?

    $a>0$. Positive $a$ makes the parabola open upward.

16. What is the vertex form of $f(x)=-x^2+6x-2$ after completing the square?

    $f(x)=-(x-3)^2+7$. Factor out $-1$: $-(x-3)^2+9-2=-(x-3)^2+7$.

17. What is the vertex form of $f(x)=x^2-8x+7$ after completing the square?

    $f(x)=(x-4)^2-9$. Complete the square: $(x-4)^2-16+7=(x-4)^2-9$.

18. What is the minimum value of $f(x)=x^2-12x+20$?

    $-16$. The $k$ value in vertex form gives the extremum.

19. What is the vertex form of $f(x)=-x^2+2x+8$ after completing the square?

    $f(x)=-(x-1)^2+9$. Factor out $-1$: $-(x-1)^2+1+8=-(x-1)^2+9$.

20. What is the vertex $y$-coordinate for $f(x)=ax^2+bx+c$ using substitution?

    $f\left(-\frac{b}{2a}\right)$. Substitute the vertex $x$-coordinate into the function.

21. What is the vertex of $f(x)=a(x-h)^2+k$ written in vertex form?

    $(h,k)$. The vertex coordinates are directly read from the form.

22. What is the minimum value of $f(x)=\frac{1}{2}x^2+3x+1$?

    $-\frac{7}{2}$. The $k$ value in vertex form gives the extremum.

23. What is the vertex form of $f(x)=x^2+10x+16$ after completing the square?

    $f(x)=(x+5)^2-9$. Complete the square: $(x+5)^2-25+16=(x+5)^2-9$.

24. What is the minimum value of $f(x)=x^2+6x+5$ found by completing the square?

    $-4$. The $k$ value in vertex form gives the extremum.

25. What is the minimum value of $f(x)=x^2-8x+7$?

    $-9$. The $k$ value in vertex form gives the extremum.

26. At what $x$ does $f(x)=x^2-8x+7$ attain its minimum value?

    $x=4$. The vertex $x$-coordinate is at $x=4$.

27. What is the vertex form of $f(x)=-x^2-4x+1$ after completing the square?

    $f(x)=-(x+2)^2+5$. Factor out $-1$: $-(x+2)^2+4+1=-(x+2)^2+5$.

28. What is the completed-square form of $x^2+bx+c$?

    $\left(x+\frac{b}{2}\right)^2+\left(c-\frac{b^2}{4}\right)$. Add and subtract $\left(\frac{b}{2}\right)^2$ to complete the square.

29. What is the vertex $x$-coordinate for $f(x)=ax^2+bx+c$ without completing the square?

    $x=-\frac{b}{2a}$. Use the vertex formula without completing the square.

30. What is the vertex form of $f(x)=x^2-2x-3$ after completing the square?

    $f(x)=(x-1)^2-4$. Complete the square: $(x-1)^2-1-3=(x-1)^2-4$.

31. What is the vertex form of $f(x)=\frac{1}{2}x^2+3x+1$ after completing the square?

    $f(x)=\frac{1}{2}(x+3)^2-\frac{7}{2}$. Factor out $\frac{1}{2}$: $\frac{1}{2}[(x+3)^2-9]+1=\frac{1}{2}(x+3)^2-\frac{7}{2}$.

32. What is the vertex form of $f(x)=2x^2-6x-8$ after completing the square?

    $f(x)=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}$. Factor out $2$: $2\left[\left(x-\frac{3}{2}\right)^2-\frac{9}{4}\right]-8=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}$.

33. What is the $x$-coordinate where $f(x)=a(x-h)^2+k$ reaches its extremum?

    $x=h$. The extremum occurs at the vertex's $x$-coordinate.

34. What is the vertex form of $f(x)=3x^2-12x+5$ after completing the square?

    $f(x)=3(x-2)^2-7$. Factor out $3$: $3[(x-2)^2-4]+5=3(x-2)^2-7$.

35. What is the vertex form of $f(x)=x^2+6x+5$ after completing the square?

    $f(x)=(x+3)^2-4$. Complete the square: $(x+3)^2-9+5=(x+3)^2-4$.

36. What is the maximum value of $f(x)=a(x-h)^2+k$ when $a<0$?

    $k$. The $y$-coordinate of the vertex is the maximum.

37. What is the axis of symmetry of $f(x)=a(x-h)^2+k$ in vertex form?

    $x=h$. The axis passes through the vertex at $x=h$.

38. What is the vertex form of $f(x)=-2x^2+4x+9$ after completing the square?

    $f(x)=-2(x-1)^2+11$. Factor out $-2$: $-2[(x-1)^2-1]+9=-2(x-1)^2+11$.

39. What is the maximum value of $f(x)=-3x^2-12x+6$?

    $18$. The $k$ value in vertex form gives the extremum.

40. What is the minimum value of $f(x)=2x^2+8x+1$?

    $-7$. The $k$ value in vertex form gives the extremum.

41. What is the maximum value of $f(x)=-x^2+6x-2$?

    $7$. The $k$ value in vertex form gives the extremum.

42. What perfect-square trinomial results from $x^2+bx+\left(\frac{b}{2}\right)^2$?

    $\left(x+\frac{b}{2}\right)^2$. This forms a perfect square trinomial.

43. What is the minimum value of $f(x)=2x^2-6x-8$?

    $-\frac{25}{2}$. The $k$ value in vertex form gives the extremum.

44. At what $x$ does $f(x)=-x^2+6x-2$ attain its maximum value?

    $x=3$. The vertex $x$-coordinate is at $x=3$.

45. What is the minimum value of $f(x)=4x^2-4x+1$?

    $0$. The $k$ value in vertex form gives the extremum.

46. At what $x$ does $f(x)=2x^2+8x+1$ attain its minimum value?

    $x=-2$. The vertex $x$-coordinate is at $x=-2$.

47. What is the maximum value of $f(x)=-2x^2+4x+9$?

    $11$. The $k$ value in vertex form gives the extremum.

48. What is the vertex form of $f(x)=2x^2+8x+1$ after completing the square?

    $f(x)=2(x+2)^2-7$. Factor out $2$: $2[(x+2)^2-4]+1=2(x+2)^2-7$.

49. At what $x$ does $f(x)=x^2+6x+5$ attain its minimum value?

    $x=-3$. The vertex $x$-coordinate is at $x=-3$.

50. What is the minimum value of $f(x)=x^2+bx$ expressed in terms of $b$?

    $-\frac{b^2}{4}$. The extremum occurs when $(x+\frac{b}{2})^2=0$.

51. What is the maximum value of $f(x)=-x^2-4x+1$?

    $5$. The $k$ value in vertex form gives the extremum.

52. What is the standard completed-square (vertex) form of a quadratic function?

    $f(x)=a(x-h)^2+k$. This is the standard form with vertex at $(h,k)$.

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