Polynomial Functions: Get Stronger Key — Precalculus Practice Page Answer Keys

Quadratic Functions Solutions

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

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

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

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

5. 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].

6. 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].

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

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

9. [latex]\left\{3i\sqrt{3},-3i\sqrt{3}\right\}[/latex]

10. [latex]\left\{2+i,2-i\right\}[/latex]

11. [latex]\left\{5+i,5-i\right\}[/latex]

12. [latex]\left\{-\frac{1}{2}+\frac{3}{2}i, -\frac{1}{2}-\frac{3}{2}i\right\}[/latex]

13. [latex]\left\{-\frac{3}{5}+\frac{1}{5}i, -\frac{3}{5}-\frac{1}{5}i\right\}[/latex]

14. 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].

Graph of f(x) = x^2-2x

15. 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].

Graph of f(x)x^2-5x-6

16. 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].

Graph of f(x)=-2x^2+5x-8

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

18. [latex]f\left(x\right)=-2{x}^{2}+8x - 1[/latex]

19. [latex]f\left(x\right)=\frac{1}{2}{x}^{2}-3x+\frac{7}{2}[/latex]

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

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

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

23. 50 feet by 50 feet. Maximize [latex]f\left(x\right)=-{x}^{2}+100x[/latex].

24. 2909.56 meters

25. $10.70

Polynomial Functions Solutions

1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

2. As x decreases without bound, so does [latex]f\left(x\right)[/latex]. As x increases without bound, so does [latex]f\left(x\right)[/latex].

3. Degree = 2, Coefficient = –2

4. Degree =4, Coefficient = –2

5. [latex]\text{As }x\to \infty ,f\left(x\right)\to \infty ,\text{ as }x\to -\infty ,f\left(x\right)\to \infty[/latex]

6. [latex]\text{As }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty[/latex]

7. [latex]\text{As }x\to \infty ,f\left(x\right)\to \infty ,\text{ as }x\to -\infty ,f\left(x\right)\to -\infty[/latex]

8. y-intercept is [latex]\left(0,12\right)[/latex], t-intercepts are [latex]\left(1,0\right);\left(-2,0\right);\text{and }\left(3,0\right)[/latex].

9. y-intercept is [latex]\left(0,-16\right)[/latex]. x-intercepts are [latex]\left(2,0\right)[/latex] and [latex]\left(-2,0\right)[/latex].

10. y-intercept is [latex]\left(0,0\right)[/latex].i x-intercepts are [latex]\left(0,0\right),\left(4,0\right)[/latex], and [latex]\left(-2, 0\right)[/latex].

11. 3

12. 5

13. [latex]\text{As }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex]

x f(x)
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

14. [latex]\text{As }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty[/latex]

x f(x)
10 –504
100 –941,094
–10 1,716
–100 1,061,106

15. The y-intercept is [latex]\left(0, 0\right)[/latex]. The x-intercepts are [latex]\left(0, 0\right),\text{ }\left(2, 0\right)[/latex]. [latex]\text{As }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex]
Graph of f(x)=x^3(x-2).

16. The y-intercept is [latex]\left(0, -81\right)[/latex]. The x-intercept are [latex]\left(3, 0\right),\text{ }\left(-3, 0\right)[/latex]. [latex]\text{As }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex]
Graph of f(x)=x^3-27.

17. The y-intercept is [latex]\left(0, 0\right)[/latex]. The x-intercepts are [latex]\left(-3, 0\right),\text{ }\left(0, 0\right),\text{ }\left(5, 0\right)[/latex]. [latex]\text{As }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex]
Graph of f(x)=-x^3+x^2+2x.

18. [latex]f\left(x\right)={x}^{2}-4[/latex]

19. [latex]f\left(x\right)={x}^{3}-4{x}^{2}+4x[/latex]

20. [latex]f\left(x\right)={x}^{4}+1[/latex]

21. [latex]V\left(x\right)=4{x}^{3}-32{x}^{2}+64x[/latex]

Graphs of Polynomial Functions Solutions

1. The x-intercept is where the graph of the function crosses the x-axis, and the zero of the function is the input value for which [latex]f\left(x\right)=0[/latex].

2. [latex]\left(-2,0\right),\left(3,0\right),\left(-5,0\right)[/latex]

3. [latex]\left(3,0\right),\left(-1,0\right),\left(0,0\right)[/latex]

4. [latex]\left(0,0\right),\text{ }\left(-5,0\right),\text{ }\left(4,0\right)[/latex]

5. [latex]\left(-2,0\right),\left(2,0\right),\left(\frac{1}{2},0\right)[/latex]

6. [latex]\left(1,0\right),\text{ }\left(-1,0\right)[/latex]

7. [latex]\left(0,0\right),\left(\sqrt{3},0\right),\left(-\sqrt{3},0\right)[/latex]

8. [latex]f\left(2\right)=-10[/latex] and [latex]f\left(4\right)=28[/latex]. Sign change confirms.

9. [latex]f\left(1\right)=3[/latex] and [latex]f\left(3\right)=-77[/latex]. Sign change confirms.

10. 0 with multiplicity 2, [latex]-\frac{3}{2}[/latex] with multiplicity 5, 4 with multiplicity 2

11. 0 with multiplicity 2, –2 with multiplicity 2

12. [latex]\text{0}\text{ with multiplicity }4\text{,}2\text{ with multiplicity }1\text{,}-\text{1}\text{ with multiplicity }1[/latex]

13. [latex]\frac{3}{2}[/latex] with multiplicity 2, 0 with multiplicity 3

14. x-intercepts, [latex]\left(1, 0\right)[/latex] with multiplicity 2, [latex]\left(-4, 0\right)[/latex] with multiplicity 1, y-intercept [latex]\left(0, 4\right)[/latex]. As [latex]x\to -\infty[/latex] , [latex]f\left(x\right)\to -\infty[/latex] , as [latex]x\to \infty[/latex] , [latex]f\left(x\right)\to \infty[/latex] .
Graph of g(x)=(x+4)(x-1)^2.

15. x-intercepts [latex]\left(3,0\right)[/latex] with multiplicity 3, [latex]\left(2,0\right)[/latex] with multiplicity 2, y-intercept [latex]\left(0,-108\right)[/latex] . As [latex]x\to -\infty[/latex], [latex]f\left(x\right)\to -\infty[/latex] , as [latex]x\to \infty[/latex] , [latex]f\left(x\right)\to \infty[/latex].
Graph of k(x)=(x-3)^3(x-2)^2.

16. x-intercepts [latex]\left(0, 0\right),\left(-2, 0\right),\left(4, 0\right)[/latex] with multiplicity 1, y-intercept [latex]\left(0, 0\right)[/latex]. As [latex]x\to -\infty[/latex] , [latex]f\left(x\right)\to \infty[/latex] , as [latex]x\to \infty[/latex] , [latex]f\left(x\right)\to -\infty[/latex].
Graph of n(x)=-3x(x+2)(x-4).

17. [latex]f\left(x\right)=-\frac{2}{9}\left(x - 3\right)\left(x+1\right)\left(x+3\right)[/latex]

18. [latex]f\left(x\right)=-\frac{1}{8}\left(x +4\right)\left(x+2\right)\left(x-1\right)\left(x-3\right)[/latex]

19. [latex]f\left(x\right)=\frac{1}{12}\left(x +2\right)^2\left(x+3\right)^2[/latex]

20. [latex]f\left(x\right)=\frac{1}{3}{\left(x - 3\right)}^{2}{\left(x - 1\right)}^{2}\left(x+3\right)[/latex]

21. [latex]f\left(x\right)=-2\left(x+3\right)\left(x+2\right)\left(x - 1\right)[/latex]

22. [latex]f\left(x\right)=-\frac{3}{2}{\left(2x - 1\right)}^{2}\left(x - 6\right)\left(x+2\right)[/latex]

23. [latex]f\left(x\right)=4{x}^{3}-36{x}^{2}+80x[/latex]

24. [latex]f\left(x\right)=4{x}^{3}-36{x}^{2}+60x+100[/latex]

25. [latex]f\left(x\right)=\pi \left(9{x}^{3}+45{x}^{2}+72x+36\right)[/latex]