Question

Select the correct answer Which table represents the increasing linear function with the greatest unit rate

Answer 1

Step-by-step explanation:

A polynomial function of degree n in the standard form is expressed as

           $f(x) \ = \ a_{n}x^n \ + \ a_{n-1}x^{n-1} \ + \ a_{n-2}x^{n-2} \ + \ a_{n-3}x^{n-3} \ + \ \cdots \ + \ \\ \\ \-\hspace{1.37cm} a_{3}x^{3} \ + \ a_{2}x^{2} \ + \ a_{1}x \ + \ a_{0}$,

where  $a_{n}, \ a_{n-1}, \ a_{n-2}, \ \cdots, \ a_{2}, \ a_{1}, \ a_{0}$  are the coefficients of the polynomial.

* The degree of a polynomial function is the highest power of which the

  variable is raised to.

A linear function is a polynomial function of degree 1 in the form

                                                $f(x) \ = \ a_{1}x \ + \ a_{0}$,

and is defined geometrically as a straight line where $a_{1}$ is the slope of the line and $a_{0}$ is the y-intercept of the function.

The slope of a linear function, $f(x)$ , measures the constant rate of change of  $f(x)$ per unit change in $x$. In other words, the steepness of the line.

Suppose that the linear function $f(x)$ contains two distinct points $(x_{1}, \ f(x_{1}))$ and $(x_{2}, \ f(x_{2}))$, then the slope of line defined by the function f is

                                       $\text{slope} \ = \ \displaystyle\frac{f(x_{2}) \ - \ f(x_{1})}{x_{2} \ - \ x_{1}}$.

A linear function can be defined as increasing, decreasing, or constant.

• It is stated that a linear function is increasing if $f(x_{2}) \ > \ f(x_{1})$ for all points $x_{1}$ and $x_{2}$ in its domain such that $x_{2} \ > \ x_{1}$. In other words, the slope of the function is positive.

• A linear function is decreasing if $f(x_{2}) \ < \ f(x_{1})$ for all points $x_{1}$ and $x_{2}$ in its domain such that $x_{2} \ < \ x_{1}$. In other words, the slope of the function is negative.

• A linear function is decreasing if $f(x_{2}) \ < \ f(x_{1})$ for all points $x_{1}$ and $x_{2}$ in its domain such that $x_{2} \ < \ x_{1}$. In other words, the slope of the function is negative.

• A linear function is constant if

1. $f(x_{2}) \ = \ f(x_{1})$ for all points $x_{1}$ and $x_{2}$ in its domain such that $x_{2} \ > \ x_{1}$. In other words, the slope of the function is zero and isdescribed geometrically as a horizontal line.
2. $f(x_{2}) \ > \ f(x_{1})$ for all points $x_{1}$ and $x_{2}$ in its domain such that $x_{2} \ = \ x_{1}$. In other words, the slope of the function is undefined and is described geometrically as a vertical line.

Given that the linear function of interest is not only an increasing function but also with the greatest unit rate, specifically, the linear function has a positive slope. Furthermore, the magnitude of the slope must also be the greatest.

Now, consider every option to the description of an increasing linear function stated above.

For option A, as x increases, y decreases. Hence, we know that the function has a negative slope.

For option B, similar to option A, as x increases, y decreases. Hence, we know that the function also has a negative slope.

For option C, it is observed that y increases as x increases. Hence, the function has a positive slope. The magnitude of the slope is therefore

                                $\text{slope(option C)} \ = \ \displaystyle\frac{-15 \ - \ (-24)}{5 \ - \ 2} \\ \\ \-\hspace{2.5cm} = \ \displaystyle\frac{9}{3} \\ \\ \-\hspace{2.5cm} = 3$

This calculation can be stated descriptively as when x increases by 3 units (from 2 to 5), y increases by 9 units (from -24 to -15). Then, the unit rate is 3 units (y increases by 3 units for every 1 unit increase in x).

For option D, similar to option A and B, where y decreases as x increases. Thus, the slope of the function is negative.

For option E, as x increases, y also increases. This implies that the function has a positive slope. The magnitude of the slope is

                                      $\text{slope(option E)} \ = \ \displaystyle\frac{-17 \ - \ (-19)}{6 \ - \ 2} \\ \\ \-\hspace{2.5cm} = \ \displaystyle\frac{2}{4} \\ \\ \-\hspace{2.5cm} = \ \displaystyle\frac{1}{2}$,

implying that y increases by 0.5 units for every 1 unit increase in x.

Therefore, the linear function of interest is of option C.