Chapter 8.3, Problem 20E

To determine

To describe: The error in graphing the function $f\left(x\right)=x^2+4x+3$ and correct it.

Answer to Problem 20E

The error is that the formula used to find the axis of symmetry is wrong,

Correct graph is:

  

Explanation of Solution

Given information:

The function:

  $f\left(x\right)=x^2+4x+3$ ….. (1)

Concept Used:

Axis of symmetry of the function $f\left(x\right)=a{x}^2+bx+c$ is given by:

  $x=-\frac{b}{2a}$

Calculation:

The error is that the formula used to find the axis of symmetry is wrong,

The formula used is $x=\frac{b}{2a}$ whereas the correct formula is $x=-\frac{b}{2a}$ .

Comparing the given function $f\left(x\right)=x^2+4x+3$ with $f\left(x\right)=a{x}^2+bx+c$ , we have that

  $a=1$ , $b=4$ And $c=3$

First find the axis of symmetry of the given parabola:

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{4}{2\left(1\right)}\\ =-2\end{array}$

So, the x -coordinate of the vertex is -2. Now use the given function to find the y -coordinate of the vertex.

  $\begin{array}{c}f\left(-2\right)={\left(-2\right)}^2+4\left(-2\right)+3\\ =4-8+3\\ =-1\end{array}$

So, the vertex is $\left(-2,-1\right)$

The y -intercept of the graph of the given function is given by the constant term of the function.

Because $c=3$ , so the y -intercept is 3

Thus, the point $\left(0,3\right)$ lies on the graph.

Since $x=-2$ is the axis of symmetry, the point $\left(-4,3\right)$ also lies on the graph of the given function.

Now, draw a smooth curve passing through the points $\left(-2,-1\right)$ , $\left(0,3\right)$ and $\left(-4,3\right)$ .

The graph of the given function is: