Question

1. $f(x) = -x^2 + 6x - 9$
2. $f(x) = -x^2 + 8x - 18$
3. $f(x) = x^5 - 3x^3 + x - 1$
4. $f(x) = x^4 - 3x^2 - x + 4$

Explanation:

Since the problem is not specified (e.g., finding vertex, roots, analyzing the function, etc.), let's assume we are to find the vertex of the quadratic functions (17 and 19) using the vertex formula. For a quadratic function \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex is \( x = -\frac{b}{2a} \), and the y - coordinate is \( f(-\frac{b}{2a}) \).

Step 1: Analyze \( f(x)=-x^{2}+6x - 9 \) (Problem 17)

For \( f(x)=-x^{2}+6x - 9 \), \( a=-1 \), \( b = 6 \), \( c=-9 \).
The x - coordinate of the vertex: \( x=-\frac{6}{2\times(-1)}=\frac{-6}{-2} = 3 \)
Substitute \( x = 3 \) into the function: \( f(3)=-(3)^{2}+6\times3 - 9=-9 + 18-9=0 \)
So the vertex of \( f(x)=-x^{2}+6x - 9 \) is \( (3,0) \)

Step 2: Analyze \( f(x)=-x^{2}+8x - 18 \) (Problem 19)

For \( f(x)=-x^{2}+8x - 18 \), \( a=-1 \), \( b = 8 \), \( c=-18 \)
The x - coordinate of the vertex: \( x=-\frac{8}{2\times(-1)}=\frac{-8}{-2}=4 \)
Substitute \( x = 4 \) into the function: \( f(4)=-(4)^{2}+8\times4-18=-16 + 32-18=-2 \)
So the vertex of \( f(x)=-x^{2}+8x - 18 \) is \( (4,-2) \)

For the polynomial functions (21 and 23), if we were to find roots or analyze end - behavior:

• For \( f(x)=x^{5}-3x^{3}+x - 1 \) (Problem 21), the degree is 5 (odd) and the leading coefficient is 1 (positive). As \( x

ightarrow+\infty \), \( f(x)
ightarrow+\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)

• For \( f(x)=x^{4}-3x^{2}-x + 4 \) (Problem 23), the degree is 4 (even) and the leading coefficient is 1 (positive). As \( x

ightarrow\pm\infty \), \( f(x)
ightarrow+\infty \)

Answer:

Since the problem is not specified, here are some possible results:

• Vertex of \( f(x)=-x^{2}+6x - 9 \): \( (3,0) \)
• Vertex of \( f(x)=-x^{2}+8x - 18 \): \( (4,-2) \)
• End - behavior of \( f(x)=x^{5}-3x^{3}+x - 1 \): As \( x

ightarrow+\infty \), \( f(x)
ightarrow+\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)

• End - behavior of \( f(x)=x^{4}-3x^{2}-x + 4 \): As \( x

ightarrow\pm\infty \), \( f(x)
ightarrow+\infty \)

(Please specify the exact problem (e.g., find vertex, roots, analyze end - behavior, etc.) for a more targeted answer)