Determine the polynomial functions for each graph on the textbook.

$$
text{color{#4257b2}(a) Polynomial function}
$$

Because the degree of exponent function is normal number and degree is $n=4$

$$
text{color{#4257b2}(b) Another function}
$$

Because the degree of exponent function is not normal number and degree is negative number.

$$
text{color{#4257b2}(c) Another function}
$$

Because the degree of exponent function is not normal number and degree is fractional number.

$$
text{color{#4257b2}(d) Polynomial function}
$$

Because the degree of exponent function is normal number and degree is $n=3$

$$
text{color{#4257b2}(e) Polynomial function}
$$

Because the degree of exponent function is normal number and degree is $n=2$

$$
text{color{#4257b2}(f) Another function}
$$

Because the degree of exponent function is not normal number.

$text{color{#4257b2}(a) Polynomial function.}$ $text{color{#4257b2}(b) Another function.}$

$text{color{#4257b2}(c) Another function.}$ $text{color{#4257b2}(d) Polynomial function.}$

$text{color{#4257b2}(e) Polynomial function.}$ $text{color{#4257b2}(f) Another function.}$

Determine each function is polynomial function or another function.

$$
color{#4257b2}text{(a)} f(x)=2x^3+x^2-5
$$

The function has a normal exponent number, so the function is a polynomial function.

$$
color{#4257b2}text{(b)} f(x)=x^2+3x-2
$$

The function has a normal exponent number, so the function is a polynomial function.

$$
color{#4257b2}text{(c)} y=2x-7
$$

The function has a normal exponent number, so the function is a polynomial function.

$$
color{#4257b2}text{(d)} y=sqrt{x+1}
$$

The function has an unnormal exponent number$color{#4257b2}left(dfrac{1}{2}right)$, so the function is another function.

$$
color{#4257b2}text{(e)} y=dfrac{x^2-4x+1}{x+2}
$$

The function has divided by another function $color{#4257b2}x+2$, so the function is another function.

$$
color{#4257b2}text{(f)} f(x)=x(x-1)^2
$$

The function has a normal exponent number, so the function is a polynomial function.

$$
text{color{Brown}(a) Polynomial function (b) Polynomial function
\ \
(c) Polynomial function (d) Another function
\ \
(e) Another function (f) Polynomial function}
$$

We need to determine the type of function with following values of $x$ and $y$.

$$
color{#4257b2}text{(a)} x=0, 500, 1000, 1500, 2000 y=200, 225, 250, 275, 300
$$

First differences as following:

$$
begin{align*}
225-200&=25 250-225=25
\ \
275-250&=25 300-275=25
end{align*}
$$

The first differences are constant, so the linear function is the best modeling for these data.

$$
color{#4257b2}text{(b)} x=0, 1, 2, 3, 4 y=10, 25, 30, 25, 10
$$

First differences as following:

$$
begin{align*}
25-10&=15 30-25=5
\ \
25-30&=-5 10-25=-15
end{align*}
$$

The first differences are not constant, so check the second difference as follows:

$$
begin{align*}
5-15&=-10 -5-5=-10
\ \
-15-(-5)&=-10
end{align*}
$$

The second differences are constant, so the quadratic function is the best modeling for these data.

$$
color{#4257b2}text{(c)} x=1, 2, 3, 4, 5 y=200, 225, 250, 275, 300
$$

First differences as following:

$$
begin{align*}
225-200&=25 250-225=25
\ \
275-250&=25 300-275=25
end{align*}
$$

The first differences are constant, so the linear function is the best modeling for these data.

$$
color{#4257b2}text{(d)} x=0, 1, 2, 3, 4, 5, 6 y=200, 204, 232, 308, 456, 700, 1064
$$

First differences as following:

$$
begin{align*}
204-200&=4 232-204=28
\ \
308-232&=76 456-308=148
\ \
700-456&=244 1064-700=364
end{align*}
$$

The first differences are not constant, so check the second difference as follows:

$$
begin{align*}
28-4&=24 76-28=48
\ \
148-76&=72 244-148=96
\ \
364-244&=120
end{align*}
$$

The second differences are not constant, so check the ratio of $y$ values as follows:

$$
dfrac{48}{24}=2 dfrac{72}{48}=1.5
$$

$$
dfrac{96}{72}=1.3 dfrac{120}{96}=1.25
$$

The ratios of $y$ values are not equal, so the data is represent another function.

$$
text{color{Brown}(a) The linear function is the best modeling for these data.
\ \
(b) The quadratic function is the best modeling for these data.
\ \
(c) The linear function is the best modeling for these data.
\ \
(d) The data is represents another function.}
$$

Graph the function of $y=2^x$ with the domain of $0le xle3$

$$
x=0 y=2^x y=2^0 y=1 (0, 1)
$$

$$
x=1 y=2^x y=2^1 y=2 (1, 2)
$$

$$
x=2 y=2^x y=2^2 y=4 (2, 4)
$$

$$
x=3 y=2^x y=2^3 y=8 (3, 8)
$$

Graph the function for the above point.

(a) Why the person see the graph might think that represent the polynomial function?

Because it seem the quadratic function like a quadratic function.

(b) Explain why this function is not a polynomial function?

Because the degree of exponent number is not a normal number, so this is another function.

$$
text{color{Brown}(a) Because it seems the quadratic function like a quadratic function.
\ \
(b) Because the degree of exponent number is not a normal number, so this is another function.}
$$

$$
text{color{#4257b2}Draw a graph of a polynomial function that has all the following characteristics.
\ \
** $f(-3)=16 f(3)=0 f(-1)=0$
\ \
** $(y)$ intercept is equal $(2)$ ** $f(x)ge0$ when $x3$ ** The domain is set for all real number}
$$

The graph has a zero point is, $(x=3)$ so this graph is represented to the cubic function with two turning points as follows:

$$
f(x)=-0.66 (x-3)(x+1)^2
$$

There are many graphs have the same characteristics.

$$
text{color{Brown}$$f(x)=-0.66 (x-3)(x+1)^2$$}
$$

Let’s choose **The $y$ intercept is $2$**. Any polynomial function that has constant term $2$ has the $y$ intercept equal to $2$. Those can be the functions
$$
begin{align*}
y&=3x^2+5x+2,\
y&=2x^2+dfrac{4}{3}x+2,\
y&=23x^2-15x+2…
end{align*}$$
Note that all of these function have different zeros and different values for the same value of $x$, but all of them has the $y$ intercept equal to $2$.
Also, note that the first function has zero for $f(-1)=0$, the second has the value $f(-3)=16$, but none of the other functions have the same characteristics.

Before you graph a function, be sure to clear any information left on the
calculator from the last time it was used.

To graph the function $y=3x^2+5x+2$ enter the function and press GRAPH. You will see that this function has the intercept with $y$-axis at $y=2$ and its zero, the place that it intercepts the $x$ axis is at $x=-1$, as depicted in *Figure 1*.

*Figure 1.* The graph of the first function

Then graph, the function $y=2x^2+frac{4}{3}x+2$. You will see that this function has the intercept with $y$-axis at $y=2$ but does not intercept the $x$ axis at all. While this function satisfies $f(-3)=16$, the previous does not, although they both intercept the $y$ axis at the same point $2$.

*Figure 2.* The graph of the second function

Create a linear, quadratic, cubic and quartic function that all have the same $y$ intercept $5$

$$
text{Linear function} f(x)=x+5
$$

$$
text{Quadratic function} f(x)=(x+1)(x+5)
$$

$$
text{Cubic function} f(x)=(x-2)(x+3)(x+4)
$$

$$
text{Quartic function} f(x)=(x+4)(x-8)(x+5)(x+4)
$$

$$
text{color{Brown}(a) $f(x)=x+5$
\ \
(b) $f(x)=(x+1)(x+5)$
\ \
(c) $f(x)=(x-2)(x+3)(x+4)$
\ \
(d) $f(x)=(x+4)(x-8)(x+5)(x+4)$}
$$

Complete chart in the textbook to summarize the polynomial functions.

$$
text{color{#4257b2}Definition}
$$

The polynomial function is any function that containing the polynomial expression in one variable.

$$
text{color{#4257b2}Examples}
$$

$$
f(x)=6x^3-3x^2+4x-9
$$

$$
text{color{#4257b2}Characteristics}
$$

** The degree of the function is the heights exponent in expression.

** The domain of the polynomials is set for all real number, $[xin R]$

** The range of the polynomials is may be set for all real number or it may be have a lower bound or an upper bound but not both.

** The graph of the polynomials function do not have a vertical or horizontal asymptotes.

$$
text{color{#4257b2}Non examples}
$$

$$
f(x)=sqrt{x+5}
$$