Question

pre - calculus ii 2.2 polynomials of a higher degree
name:
match the possible equation with the given graph:

1. $f(x)=-2x + 3$
2. $f(x)=-\frac{1}{4}x^{4}+3x^{2}$
3. $f(x)=-2x^{2}-5x$
4. $f(x)=\frac{1}{5}x^{5}-2x^{3}+\frac{9}{5}x$
5. $f(x)=x^{2}-4x$
6. $f(x)=2x^{3}-3x + 1$
7. $f(x)=-\frac{1}{3}x^{3}+x^{2}-1$
8. $f(x)=x^{4}+2x^{3}$

#1 - 17 no graphing calculator

Explanation:

Step1: Analyze degree and end - behavior

For a linear function $y = ax + b$ ($a
eq0$), like $f(x)=-2x + 3$, the graph is a straight line. For a quadratic function $y=ax^{2}+bx + c$ ($a
eq0$), the graph is a parabola. Higher - degree polynomials have more complex shapes. If the leading coefficient $a$ of a polynomial $y = a_nx^n+\cdots+a_0$ ($n\geq1$) and $n$ is even, the ends of the graph both go up if $a>0$ and both go down if $a < 0$. If $n$ is odd, one end goes up and one end goes down.

Step2: Analyze number of roots and turning points

The number of roots of a polynomial $f(x)$ is related to the number of times the graph crosses or touches the $x$ - axis. The number of turning points of a polynomial of degree $n$ is at most $n - 1$.

1. $f(x)=-2x + 3$ is a linear function, so it is a straight - line graph.
2. $f(x)=-\frac{1}{4}x^{4}+3x^{2}$ is an even - degree (degree 4) polynomial with a negative leading coefficient, so the ends of the graph go down.
3. $f(x)=-2x^{2}-5x=-x(2x + 5)$ is a quadratic function with a negative leading coefficient, so it is a downward - opening parabola.
4. $f(x)=\frac{1}{5}x^{5}-2x^{3}+\frac{9}{5}x$ is an odd - degree (degree 5) polynomial.
5. $f(x)=x^{2}-4x=x(x - 4)$ is a quadratic function with a positive leading coefficient, so it is an upward - opening parabola.
6. $f(x)=2x^{3}-3x + 1$ is an odd - degree (degree 3) polynomial.
7. $f(x)=-\frac{1}{3}x^{3}+x^{2}-1$ is an odd - degree (degree 3) polynomial.
8. $f(x)=x^{4}+2x^{3}=x^{3}(x + 2)$ is an even - degree (degree 4) polynomial with a positive leading coefficient, so the ends of the graph go up.

Without seeing the actual graphs to match one - by - one, we can't give a definite match. But the general approach is as above. If we assume we are matching based on the described characteristics:

1. $f(x)=-2x + 3$ would match a straight - line graph.
2. $f(x)=-\frac{1}{4}x^{4}+3x^{2}$ would match an even - degree, downward - opening graph.
3. $f(x)=-2x^{2}-5x$ would match a downward - opening parabola.
4. $f(x)=\frac{1}{5}x^{5}-2x^{3}+\frac{9}{5}x$ would match an odd - degree, more complex graph.
5. $f(x)=x^{2}-4x$ would match an upward - opening parabola.
6. $f(x)=2x^{3}-3x + 1$ would match an odd - degree graph.
7. $f(x)=-\frac{1}{3}x^{3}+x^{2}-1$ would match an odd - degree graph.
8. $f(x)=x^{4}+2x^{3}$ would match an even - degree, upward - opening graph.

Answer:

Since no specific graph - to - function match is given, we can't provide a final answer for the match. But the analysis for each function's graph characteristics is done above.