Q. 100 - Precalculus Enhanced with Graphing Utilities

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.

Step-by-Step Solution

Answer

The function values for corresponding values of b

The new functions are

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

The nature can be given as

Step 1: Given information

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

Step 2: To find the nature substitute the values of b in the equation.

When 

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

Step 3: plot the graph of the functions

The graph can be given as

Step 4: Conclusion

The nature of the family can be given as