Question

On one set of coordinate axes, graph the family of parabolas $f\left(x\right)=x^2+bx+1$for b = -4, b = 0, and b = 4. Describe the general characteristics of this family.

Short answer

The function values for corresponding values of b

The new functions are

$$\begin{gathered} whenb=0 \\ {f}_1\left(x\right)=x^2+1 \\ whenb=4 \\ {f}_2\left(x\right)=x^2+4x+1 \\ whenb=-4 \\ {f}_3\left(x\right)=x^2-4x+1 \end{gathered}$$

The nature can be given as

Step-by-step solution

Given information

We are given a quadratic equation$f\left(x\right)=x^2+bx+1$

To find the nature substitute the values of b in the equation.

When

$$\begin{gathered} b=0weget \\ {f}_1\left(x\right)=x^2+1 \\ Whenb=4weget \\ {f}_1\left(x\right)=x^2+4x+1 \\ Whenb=-4weget \\ {f}_1\left(x\right)=x^2-4x+1 \end{gathered}$$

plot the graph of the functions 

The graph can be given as

Conclusion

The nature of the family can be given as