Chapter 4.3, Problem 27PPSE

Solution

a.

To find: The equation of parabola that contains three given points.

There is no parabola containing points $\left(\text{0,-}\text{4}\right)\text{,}\left(\text{2,4}\right)$ and $\left(4,4\right)$ .

Given data: The parabola contains points $\left(\text{0,-}\text{4}\right)\text{,}\left(\text{2,4}\right)$ and $\left(4,4\right)$ .

Method/Formula used:

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

Explanation:

Let the equation of parabola that contains points $\left(\text{0,-}\text{4}\right)\text{,}\left(\text{2,4}\right)$ and $\left(4,4\right)$ is

  $y=a{x}^2+bx+c$ … (1)

Therefore, each point should satisfy the equation (1).

Substitute $x=0,y=-4$ in equation (1),

  $\left.\begin{array}{c}-4=a{\left(0\right)}^2+b\left(0\right)+c\\ 4=0+0+c\\ \Rightarrow c=4\end{array}\right\}$ ... (2)

Substitute $x=2,y=4$ in equation (1),

  $\begin{array}{l}4=a{\left(2\right)}^2+b\left(2\right)+c\\ 4=4a+2b+c\end{array}$

OR

  $4a+2b+c=4$ ... (3)

Substitute $x=4,y=\text{4}$ in equation (1),

  $\left.\begin{array}{l}4=a{\left(4\right)}^2+b\left(4\right)+c\\ 4=16a+4b+c\end{array}\right\}$

OR

  $16a+4b+c=4$ ... (4)

Subtract equation (3) from equation (2),

  $\begin{array}{c}\left(16a+4b+c\right)-\left(4a+2b+c\right)=4-4\\ 16a-4a+4b-2b+c-c=0\\ 12a+2b=0\leftarrow \text{Divide}\text{by}\text{2}\\ 6a+b=0\end{array}$

OR

  $b=-6a$ … (5)

Substitute $-6a$ for b , 4 for c in equation (3)

  $\begin{array}{c}4a+2\left(-6a\right)+4=4\\ 4a-12a=0\\ -8a=0\\ \Rightarrow a=0\end{array}$

Substitute 0 for $a$ in equation (5),

  $\begin{array}{c}b=-6\cdot 0\\ =0\end{array}$

Substitute 0 for each $a$ and b , 4 for c in equation (1),

  $\begin{array}{c}y=0x^2+0x+4\\ =4\end{array}$

But $y=4$ is a straight line parallel to x -axis.

Therefore, there can be no parabola containing points $\left(\text{0,-}\text{4}\right)\text{,}\left(\text{2,4}\right)$ and $\left(4,4\right)$ .

b.

To explain: The minimum number of data points to find a single quadratic model,

The minimum number of data points needed to find a single quadratic model is 3.

Given:

A single quadratic model is required.

Method used:

The quadric function $y=a{x}^2+bx+c$ where $a,b,c$ are constant is called quadratic model.

Explanation:

The quadric function $y=a{x}^2+bx+c$ where $a,b,c$ are constant is the quadratic model.

Since, the quadratic model $y=a{x}^2+bx+c$ has 3 unknowns $a,b,c$ so to determine those 3 linear equations in x and y are required. The required 3 equations can only be obtained if graph of $y=a{x}^2+bx+c$ contains 3 points.