Question

directions: determine the end - behavior for the following polynomials.

1. (f(x)=-4x^{3}) left: right:
2. (g(x)=3x^{6}) left: right:
3. (y = 3(x - 1)^{5}) left: right:
4. (h(x)=8 - 3x^{4}) left: right:
5. (k(x)=-8x^{2}+4 - x^{5}) left: right:
6. (m(x)=2x(x - 1)(x + 6)) left: right:
7. (p(x)=-2x(x - 3)^{2}) left: right:
8. the graphs, equations, and limit statements for four polynomial functions are below. match the graphs and equations.

Explanation:

Step1: Recall end - behavior rule

For a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_0\), the end - behavior is determined by the leading term \(a_nx^n\). If \(n\) is even and \(a_n>0\), as \(x\to\pm\infty\), \(y\to+\infty\). If \(n\) is even and \(a_n < 0\), as \(x\to\pm\infty\), \(y\to-\infty\). If \(n\) is odd and \(a_n>0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\). If \(n\) is odd and \(a_n < 0\), as \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to-\infty\).

Step2: Analyze \(f(x)=- 4x^3\)

The degree \(n = 3\) (odd) and \(a=-4<0\). As \(x\to-\infty\), \(y\to+\infty\) (Left: Up) and as \(x\to+\infty\), \(y\to-\infty\) (Right: Down).

Step3: Analyze \(g(x)=3x^6\)

The degree \(n = 6\) (even) and \(a = 3>0\). As \(x\to-\infty\), \(y\to+\infty\) (Left: Up) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step4: Analyze \(y = 3(x - 1)^5\)

The degree \(n = 5\) (odd) and \(a = 3>0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step5: Analyze \(h(x)=8-3x^4\)

Rewrite as \(h(x)=-3x^4 + 8\). The degree \(n = 4\) (even) and \(a=-3<0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to-\infty\) (Right: Down).

Step6: Analyze \(k(x)=-8x^2+x^5\)

The leading term is \(x^5\) (degree \(n = 5\), odd) and \(a = 1>0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step7: Analyze \(m(x)=2x(x - 1)(x + 6)\)

Expand to get \(m(x)=2x(x^2+5x - 6)=2x^3+10x^2-12x\). The leading term is \(2x^3\) (degree \(n = 3\), odd) and \(a = 2>0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step8: Analyze \(p(x)=-2x(x - 3)^2\)

Expand \((x - 3)^2=x^2-6x + 9\), then \(p(x)=-2x(x^2-6x + 9)=-2x^3 + 12x^2-18x\). The leading term is \(-2x^3\) (degree \(n = 3\), odd) and \(a=-2<0\). As \(x\to-\infty\), \(y\to+\infty\) (Left: Up) and as \(x\to+\infty\), \(y\to-\infty\) (Right: Down).

1. Left: Up, Right: Down
2. Left: Up, Right: Up
3. Left: Down, Right: Up
4. Left: Down, Right: Down
5. Left: Down, Right: Up
6. Left: Down, Right: Up
7. Left: Up, Right: Down

Answer:

Step1: Recall end - behavior rule

For a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_0\), the end - behavior is determined by the leading term \(a_nx^n\). If \(n\) is even and \(a_n>0\), as \(x\to\pm\infty\), \(y\to+\infty\). If \(n\) is even and \(a_n < 0\), as \(x\to\pm\infty\), \(y\to-\infty\). If \(n\) is odd and \(a_n>0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\). If \(n\) is odd and \(a_n < 0\), as \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to-\infty\).

Step2: Analyze \(f(x)=- 4x^3\)

The degree \(n = 3\) (odd) and \(a=-4<0\). As \(x\to-\infty\), \(y\to+\infty\) (Left: Up) and as \(x\to+\infty\), \(y\to-\infty\) (Right: Down).

Step3: Analyze \(g(x)=3x^6\)

The degree \(n = 6\) (even) and \(a = 3>0\). As \(x\to-\infty\), \(y\to+\infty\) (Left: Up) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step4: Analyze \(y = 3(x - 1)^5\)

The degree \(n = 5\) (odd) and \(a = 3>0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step5: Analyze \(h(x)=8-3x^4\)

Rewrite as \(h(x)=-3x^4 + 8\). The degree \(n = 4\) (even) and \(a=-3<0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to-\infty\) (Right: Down).

Step6: Analyze \(k(x)=-8x^2+x^5\)

The leading term is \(x^5\) (degree \(n = 5\), odd) and \(a = 1>0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step7: Analyze \(m(x)=2x(x - 1)(x + 6)\)

Expand to get \(m(x)=2x(x^2+5x - 6)=2x^3+10x^2-12x\). The leading term is \(2x^3\) (degree \(n = 3\), odd) and \(a = 2>0\). As \(x\to-\infty\), \(y\to-\infty\) (Left: Down) and as \(x\to+\infty\), \(y\to+\infty\) (Right: Up).

Step8: Analyze \(p(x)=-2x(x - 3)^2\)

Expand \((x - 3)^2=x^2-6x + 9\), then \(p(x)=-2x(x^2-6x + 9)=-2x^3 + 12x^2-18x\). The leading term is \(-2x^3\) (degree \(n = 3\), odd) and \(a=-2<0\). As \(x\to-\infty\), \(y\to+\infty\) (Left: Up) and as \(x\to+\infty\), \(y\to-\infty\) (Right: Down).

1. Left: Up, Right: Down
2. Left: Up, Right: Up
3. Left: Down, Right: Up
4. Left: Down, Right: Down
5. Left: Down, Right: Up
6. Left: Down, Right: Up
7. Left: Up, Right: Down