Question

Which type of function best models the data in the table? Justify your choice.

Answer 1

Answer:

Quadratic equation:  $y=-2x^2+32x+22$

Explanation:

We need to find best model of given data in the table.

1) Linear: For linear slope of the line must be same. Slope is change in y over change x must be same. But if we find change in y over change x for this table. They won't be same. This table value can't be linear.

$\frac{142-118}{6-4}\neq \frac{150-142}{8-6}$

2) Quadratic: For quadratic equation. we will take standard form of quadratic equation $y=ax^2+bx+c$. Using any three value of table we will find a, b and c and then we satisfy other value of of table.

$y=ax^2+bx+c$

Point 1: (4,118) $\Rightarrow 118=16a+4b+c$

Point 2: (6,142) $\Rightarrow 142=36a+6b+c$

Point 3: (8,150) $\Rightarrow 150=64a+8b+c$

Now we have three equation and to solve for a, b and c. Using elimination method we get the value of a=-2, b=32 and c=22.

So, Quadratic equation $y=-2x^2+32x+22$

So, we get quadratic model for the given data in the table.

Answer 2

The table should be modeled in the quadratic form as the values of variable x and y is not constant for the linear form.The function best models the data in the table is,

$y=-2x^2+32x+22$

How to model the function from the data table?

Table is a way to represent the data of the two or more variable. The linear or quadratic function, can be model with the data table.

• Linear model- The highest power of unknown variable in linear model is 1.To construct the linear model with the values given in the table, the slope of the two lines should be equal.

• Quadratic model- The highest power of unknown variable in linear model is 2. To construct the quadratic model, the standard form of quadratic equation must satisfy for all the of the data table.

Given information-

Variable x represent the games made (in 1000) in data table.

Variable y represent the profit (in $1000) in data table.  

Lets find the slope with the values given in the table to check whether, the model can be formed in linear form or not.

$\dfrac{150-142}{8-6}=\dfrac{142-118}{6-4}\\\dfrac{8}{2}=\dfrac{24}{2}\\4\neq12$

Hence, the model can not be formed in linear form.

Construct the model in quadratic form. The standard form of the quadratic equation is,

$y=ax^2+bx+c$

• Put the values from the data table from row 1 as,

         $116=16a+4b+c$             .....1

• Put the values from the data table from row 2 as,

         $142=36+6b+c$               ......2

• Put the values from the data table from row 3 as,

         $150=64+8b+c$               ......3

On solving the equation 1,2 and 3 we get,

$a=-2\\b=32\\c=22$

Put these values in the standard form as,

$y=-2x^2+32x+22$

Hence the function best models the data in the table is,

$y=-2x^2+32x+22$

Learn more about the data table here;

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