Q. 99 - Precalculus Enhanced with Graphing Utilities

Question

On one set of coordinate axes, graph the family of parabolas $f\left(x\right)=x^2+2x+c$ for c = -3, c = 0, and c = 1. Describe the characteristics of a member of this family.

Step-by-Step Solution

Answer

The function for corresponding values of c are

$$\begin{gathered} when c=0 \\ {f}_1\left(x\right)=x^2+2x \\ when c=1 \\ {f}_2\left(x\right)=x^2+2x+1 \\ when c=-3 \\ {f}_3\left(x\right)=x^2+2x-3 \end{gathered}$$

The nature can be given as

Step 1: Given information

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

Step 2: To find the nature Substitute the given values of c in the given equation

We have

$$\begin{gathered} f\left(x\right)=x^2+2x+c \\ When c=0 \\ {f}_1\left(x\right)=x^2+2x \\ When c=1 \\ {f}_2\left(x\right)=x^2+2x+1 \\ when c=-3 \\ {f}_3\left(x\right)=x^2+2x-3 \end{gathered}$$

Step 3: Now we plot the graph to find the nature

The graph can be given as

Step 4: Conclusion

The nature can be given as