Solutions to Odd-Numbered Exercises

1. When written in that form, the vertex can be easily identified.

3. If [latex]a=0[/latex] then the function becomes a linear function.

5. If possible, we can use factoring. Otherwise, we can use the quadratic formula.

7. [latex]f\left(x\right)={\left(x+1\right)}^{2}-2[/latex], Vertex [latex]\left(-1,-4\right)[/latex]

9. [latex]f\left(x\right)={\left(x+\frac{5}{2}\right)}^{2}-\frac{33}{4}[/latex], Vertex [latex]\left(-\frac{5}{2},-\frac{33}{4}\right)[/latex]

11. [latex]f\left(x\right)=3{\left(x - 1\right)}^{2}-12[/latex], Vertex [latex]\left(1,-12\right)[/latex]

13. [latex]f\left(x\right)=3{\left(x-\frac{5}{6}\right)}^{2}-\frac{37}{12}[/latex], Vertex [latex]\left(\frac{5}{6},-\frac{37}{12}\right)[/latex]

15. Minimum is [latex]-\frac{17}{2}[/latex] and occurs at [latex]\frac{5}{2}[/latex]. Axis of symmetry is [latex]x=\frac{5}{2}[/latex].

17. Minimum is [latex]-\frac{17}{16}[/latex] and occurs at [latex]-\frac{1}{8}[/latex]. Axis of symmetry is [latex]x=-\frac{1}{8}[/latex].

19. Minimum is [latex]-\frac{7}{2}[/latex] and occurs at –3. Axis of symmetry is [latex]x=-3[/latex].

21. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[2,\infty \right)[/latex].

23. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[-5,\infty \right)[/latex].

25. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[-12,\infty \right)[/latex].

27. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[-2,\infty \right)[/latex].

29. Domain is [latex]\left(-\infty ,\infty \right)[/latex] Range is [latex]\left(-\infty ,11\right][/latex].

31. [latex]f\left(x\right)={x}^{2}+1[/latex]

33. [latex]f\left(x\right)=\frac{6}{49}{x}^{2}+\frac{60}{49}x+\frac{297}{49}[/latex]

35. [latex]f\left(x\right)=-{x}^{2}+1[/latex]

37. Vertex [latex]\left(1,\text{ }-1\right)[/latex], Axis of symmetry is [latex]x=1[/latex]. Intercepts are [latex]\left(0,0\right), \left(2,0\right)[/latex].

39. Vertex [latex]\left(\frac{5}{2},\frac{-49}{4}\right)[/latex], Axis of symmetry is [latex]\left(0,-6\right),\left(-1,0\right),\left(6,0\right)[/latex].

41. Vertex [latex]\left(\frac{5}{4}, -\frac{39}{8}\right)[/latex], Axis of symmetry is [latex]x=\frac{5}{4}[/latex]. Intercepts are [latex]\left(0, -8\right)[/latex].

43. 50 feet

45. 50 feet by 50 feet. Maximize [latex]f\left(x\right)=-{x}^{2}+100x[/latex].

47. 125 feet by 62.5 feet. Maximize [latex]f\left(x\right)=-2{x}^{2}+250x[/latex].

49. 6 and –6; product is –36; maximize [latex]f\left(x\right)={x}^{2}+12x[/latex].

51. 2909.56 meters

53. $10.70