Chapter 7.6, Problem 8WE

To determine

Sketch the graph of the function $g\left(x\right)=x^2+4x+1$ .

Explanation of Solution

Given information: The given function is $g\left(x\right)=x^2+4x+1$ .

Graph: The given function can be written as,

  $\begin{array}{c}y=x^2+4x+\mathrm{1....}\left(\text{Replace }g\left(x\right)=y\right)\\ \text{or, }y-1=x^2+4x\\ y-1+\left(4\right)=\left(x^2+2\left(x\right)\left(2\right)+{\left(2\right)}^2\right)\mathrm{....}\left(\text{Complete square}\right)\\ y+3={\left(x+2\right)}^2\mathrm{....}\left(1\right)\end{array}$

Now (1) represents a parabola and its graph is shown as,

  

Observation: Since (1) represents a parabola of the form $y-k=a{\left(x-h\right)}^2$ . Therefore the graph for the function (1) has vertex $\left(-2,-3\right)$ and axis of symmetry $x=-2$ .