Use Factoring, Squares to Analyze Graphs - Algebra 1

1. Rewrite in vertex form by completing the square: $f(x)=-x^2+6x-2$.

   $f(x)=-(x-3)^2+7$. Factor out $-1$, then complete square inside.

2. For $h(t)=-16t^2+32t+5$, what is the maximum height?

   Maximum height is $21$. Substitute $t=1$: $h(1)=-16+32+5=21$.

3. What is the first step to complete the square for $ax^2+bx+c$ when $a\neq 1$?

   Factor out $a$ from the $x^2$ and $x$ terms. Makes coefficient of $x^2$ term equal to $1$.

4. Identify the axis of symmetry for $f(x)=(x-4)^2-9$.

   $x=4$. In vertex form, $h=4$ gives axis $x=4$.

5. Find the vertex of $f(x)=x^2-4x+9$ without graphing.

   $(2,5)$. Axis at $x=2$; substitute to get $y=5$.

6. What is the vertex form of a quadratic function?

   $f(x)=a(x-h)^2+k$. Standard vertex form where $ (h,k) $ is the vertex.

7. Identify the vertex of $f(x)=-2(x+3)^2+5$.

   $(-3,5)$. Vertex form shows $(h,k)=(-3,5)$ directly.

8. Factor $f(x)=x^2+7x+12$ to show its zeros.

   $f(x)=(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$.

9. Find the vertex of $f(x)=2x^2+12x+1$ without graphing.

   $(-3,-17)$. Axis at $x=-3$; substitute to get $y=-17$.

10. Rewrite in vertex form by completing the square: $f(x)=x^2-6x+10$.

    $f(x)=(x-3)^2+1$. Complete square: $x^2-6x=(x-3)^2-9$.

11. A ball has height $h(t)=-16t^2+32t+5$. What is the time of maximum height?

    $t=1$. Use axis formula: $t=-\frac{32}{2(-16)}=1$.

12. Factor $f(x)=2x^2+6x$ completely to show its zeros.

    $f(x)=2x(x+3)$. Factor out common factor $2x$ first.

13. Complete the square: what is $x^2+2x+7$ in vertex form?

    $(x+1)^2+6$. Complete square on $x^2+2x$, then add $6$.

14. Rewrite in vertex form by completing the square: $f(x)=x^2+4x-1$.

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

15. Complete the square: what is $x^2-8x$ rewritten as a square minus a constant?

    $(x-4)^2-16$. Add and subtract $(\frac{-8}{2})^2=16$.

16. What is the $y$-intercept of $f(x)=ax^2+bx+c$?

    $(0,c)$. Where parabola crosses $y$-axis at $x=0$.

17. Identify the axis of symmetry for $f(x)=(x-7)(x-1)$.

    $x=4$. Midpoint of zeros: $\frac{7+1}{2}=4$.

18. What number do you add and subtract to complete the square for $x^2+bx$?

    Add and subtract $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, squared.

19. What is the symmetry relationship between zeros $r_1$ and $r_2$ and the axis of symmetry?

    Axis is midpoint: $x=\frac{r_1+r_2}{2}$. Axis is equidistant from both zeros.

20. What is the extreme value of $f(x)=a(x-h)^2+k$ in terms of $k$ and $a$?

    Extreme value is $k$ (min if $a>0$, max if $a<0$). Value $k$ is minimum when $a>0$, maximum when $a<0$.

21. Evaluate the vertex $x$-coordinate for $f(x)=x^2+10x+1$.

    $x=-5$. Using formula $x=-\frac{b}{2a}=-\frac{10}{2(1)}=-5$.

22. What is the factored (intercept) form of a quadratic function with zeros $r_1$ and $r_2$?

    $f(x)=a(x-r_1)(x-r_2)$. Factored form directly shows zeros at $x=r_1$ and $x=r_2$.

23. What is the discriminant used to determine the number of real zeros of $ax^2+bx+c$?

    $\Delta=b^2-4ac$. Formula under square root in quadratic formula.

24. Find the minimum value of $f(x)=x^2-10x+30$.

    Minimum value is $5$. Complete square to find vertex $y$-coordinate.

25. Rewrite in vertex form by completing the square: $f(x)=2x^2+8x+3$.

    $f(x)=2(x+2)^2-5$. Factor out $2$, then complete square inside.

26. Find the axis of symmetry for $f(x)=2x^2-8x+1$.

    $x=2$. Using formula $x=-\frac{b}{2a}=-\frac{-8}{2(2)}=2$.

27. Identify the zeros of $f(x)=(x-2)(x+5)$.

    $x=2$ and $x=-5$. Set each factor equal to zero.

28. What is the axis of symmetry for $f(x)=a(x-h)^2+k$?

    $x=h$. Vertical line through vertex at $x=h$.

29. A model is $f(x)=a(x-r_1)(x-r_2)$ with $a>0$. What does $f(x)>0$ mean between the zeros?

    It is false; between zeros $f(x)<0$ when $a>0$. When $a>0$, parabola is below $x$-axis between zeros.

30. What does $\Delta=b^2-4ac<0$ imply about the zeros of $ax^2+bx+c$?

    No real zeros. Negative discriminant means no $x$-intercepts.

31. Find the axis of symmetry for $f(x)=-3x^2+12x-5$.

    $x=2$. Using formula $x=-\frac{b}{2a}=-\frac{12}{2(-3)}=2$.

32. A profit model is $P(x)=-x^2+10x-9$. What output does the vertex represent?

    The maximum profit. Since $a=-1<0$, vertex gives maximum profit.

33. Identify the axis of symmetry for $f(x)=x^2-2x-15$ using its factored form.

    $x=1$. First factor to find zeros, then find midpoint.

34. Find the axis of symmetry for $f(x)=x^2+6x+8$ by factoring first.

    $x=-3$. Factor first: $(x+2)(x+4)$; midpoint of $-2,-4$.

35. What does the sign of $a$ tell you about the opening of $f(x)=ax^2+bx+c$?

    $a>0$ opens up; $a<0$ opens down. Coefficient $a$ determines parabola direction.

36. Find the axis of symmetry for $f(x)=(x+1)(x+9)$.

    $x=-5$. Zeros are at $x=-1$ and $x=-9$; midpoint is $-5$.

37. What is the axis of symmetry for $f(x)=ax^2+bx+c$?

    $x=-\frac{b}{2a}$. Formula derived from vertex $x$-coordinate.

38. What is the meaning of a zero of a quadratic function in a real-world context?

    An input where the output equals $0$. Where the function value equals zero in context.

39. A revenue model is $R(x)=x^2-12x+40$. What does the vertex represent?

    The minimum revenue. Since $a=1>0$, vertex gives minimum revenue.

40. A projectile has $h(t)=-5t^2+20t$. What are the zeros interpreted as in context?

    Times when height is $0$ (ground level). Zeros represent when projectile hits ground.

41. Find the maximum value of $f(x)=-2x^2+8x+1$.

    Maximum value is $9$. Since $a=-2<0$, vertex gives maximum.

42. Complete the square: what is $x^2+6x$ rewritten as a square minus a constant?

    $(x+3)^2-9$. Add and subtract $(\frac{6}{2})^2=9$.

43. What is the extreme value of $f(x)=3(x-1)^2-7$?

    Minimum value is $-7$. Since $a=3>0$, minimum occurs at $k=-7$.

44. Identify the zeros of $f(x)=2x(x-5)$.

    $x=0$ and $x=5$. Set each factor equal to zero and solve.

45. Factor $f(x)=x^2-9$ to show its zeros.

    $f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.

46. What is the minimum or maximum value of $f(x)$ called in a context problem?

    The extreme value (minimum or maximum output). The optimal value in optimization problems.

47. What are the zeros of $f(x)=a(x-r_1)(x-r_2)$?

    $x=r_1$ and $x=r_2$. Values that make each factor equal zero.

48. Factor $f(x)=x^2-5x+6$ to show its zeros.

    $f(x)=(x-2)(x-3)$. Find two numbers that multiply to $6$ and add to $-5$.

49. What is the meaning of the axis of symmetry in a real-world context?

    The input where the extreme value occurs. The optimal input value for extreme output.

50. What does $\Delta=b^2-4ac=0$ imply about the zeros of $ax^2+bx+c$?

    One real double zero. Zero discriminant gives one $x$-intercept (vertex on axis).

51. Find the vertex of $f(x)=(x-1)(x-5)$ using symmetry and substitution.

    $(3,-4)$. Zeros at $x=1,5$; vertex at midpoint $x=3$.

52. What perfect-square trinomial results from completing the square on $x^2+bx$?

    $\left(x+\frac{b}{2}\right)^2$. Perfect square from completing the square process.

53. What does $\Delta=b^2-4ac>0$ imply about the zeros of $ax^2+bx+c$?

    Two distinct real zeros. Positive discriminant gives two $x$-intercepts.

54. Complete the square: what is $x^2-12x+20$ in vertex form?

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

55. What is the extreme value of $f(x)=-\frac{1}{2}(x-6)^2+4$?

    Maximum value is $4$. Since $a=-\frac{1}{2}<0$, maximum occurs at $k=4$.

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