We are asked to determine polynomic functions which graph resembles the given letters.
For the letter U we will use a second-degree polynomial, which means a polynomial of the form:
[tex]y=ax^2+b[/tex]
Is we take the values of "a" and "b" to be:
[tex]\begin{gathered} a=1 \\ b=0 \end{gathered}[/tex]
We get the function:
[tex]y=x{}^2[/tex]
The graph is the following:
Now, to determine a function that resembles the letter N we will use a polynomic function of third-degree, this means a function of the form:
[tex]y=ax^3+bx^2+cx+d[/tex]
We will use the following values for the constants:
[tex]\begin{gathered} a=\frac{1}{4} \\ \\ b=1 \\ c=0 \\ d=0 \end{gathered}[/tex]
Substituting we get:
[tex]y=\frac{1}{4}x{}^3+x^2[/tex]
The graph of the function is:
To determine a polynomial that resembles the letter "m" we will use a polynomial that has 3 x-intercepts and the end-points are pointing down. This means that the function is of the form:
[tex]y=-(x-a)(x-b)^2(x-c)[/tex]
The middle term has a square because we want the middle intercept to be tangent to the x-axis. Giving values to the constant we get:
[tex]y=-(x+1)(x-1)^2(x-3)[/tex]
The graph of the function is:
Now, we determine a function that resembles the letter "W". We will use a polynomial with two intercepts that are tangent to the x-axis and the end behavior must be upwards. Therefore, the function must be of the form:
[tex]y=(x-a)^2(x-b)^2[/tex]
We will use a = -1 and b = 1:
[tex]y=(x+1)^2(x-1)^2[/tex]
The graph is:
