Question

Function I is defined by the graph, function II is defined by the table, and function III is defined by the equation $y=4.5x+9.$


$\begin{array}{cc}x& y\\ 0& 9\\ 1& 14\\ 2& 19\\ 3& 24\\ 4& 29\end{array}$

The given options represent the functions in decreasing order based on their slopes. Choose the option correctly representing the order.

Answer 1

Detailed step-by-step solution:
Finding the slope of function I:
Consider two points $(-1,-1)$ and $(0,3)$ on the graph.
${\text{Slope}}_I=\frac{\text{Rise}}{\text{Run}}$
$=\frac{\text{Change in y}}{\text{Change in y}}$
$=\frac{3-(-1)}{0-(-1)}$
$=\frac{3+1}{1}$
$=4$
Finding the slope of function II:
Consider two ordered pairs $(1,14)$ and $(2,19)$ from the table.
${\text{Slope}}_{II}=\frac{\text{Rise}}{\text{Run}}$
$=\frac{\text{Change in y}}{\text{Change in y}}$
$=\frac{19-14}{2-1}$
$=\frac{5}{1}$
$=5$
Finding the slope of function III: $y=4.5x+9$
For the equation $y=mx+b$
$\Rightarrow m=$ slope
Comparing $y=4.5x+9$ with $y=mx+b,$ we get:
${\text{Slope}}_{III}=m=4.5$
$5>4.5>4$
$\Rightarrow {\text{Slope}}_{II}>{\text{Slope}}_{III}>{\text{Slope}}_I$
Decreasing order of the functions based on their slopes will be:
$\Rightarrow$ Function II $>$ Function III $>$ Function I
$\Rightarrow$ Option A is correct.