Chapter 2.1, Problem 35E

To determine

Determine the standard form of the quadratic function by using the graph below.

  

Answer to Problem 35E

  $y=x^2+4x+3$

Explanation of Solution

Given:

  

Calculation:

The standard form of quadratic function is $y=a{x}^2+bx+c$ .

From graph, it is clear that, the function (curve) passes through the points

  $(0,3),(-3,0)\text{and}(-2,-1).$

Now, substitute these points in the standard form.

  $\begin{array}{l}i)(0,3)\\ \Rightarrow y=a{x}^2+bx+c\\ $ \Rightarrow 3=a{(0)}^2+b(0)+c$\Rightarrow 3=a(0)+b(0)+c\\ \Rightarrow 3=0+0+c\\ \Rightarrow 3=c\\ \Rightarrow c=3\\ \\ ii)(-3,0)\\ \Rightarrow y=a{x}^2+bx+c\\ $ \Rightarrow 0=a{(-3)}^2+b(-3)+c$\Rightarrow 0=a(9)+b(-3)+c\\ \Rightarrow 0=9a-3b+3[\because c=3]\\ \Rightarrow -3=9a-3b\\ \Rightarrow 9a-3b=-\mathrm{3.....}(i)\\ \\ iii)(-2,-1)\\ \Rightarrow y=a{x}^2+bx+c\\ $ \Rightarrow -1=a{(-2)}^2+b(-2)+c$\Rightarrow -1=a(4)+b(-2)+c\\ \Rightarrow -1=4a-2b+3[\because c=3]\\ \Rightarrow -1-3=4a-2b\\ \Rightarrow 4a-2b=-\mathrm{4.....}(ii)\end{array}$

By solving equation (i) and equation (ii), we get

  $a=1\text{and}\text{}b=4$

Substitute values of a , b and c in standard form of quadratic function.

  $\begin{array}{l}\Rightarrow y=a{x}^2+bx+c\\ \Rightarrow y=(1)x^2+(4)x+3\\ \Rightarrow y=x^2+4x+3\end{array}$

Conclusion:

Therefore, the standard form is $y=x^2+4x+3$ .