Question

The function f(x) is shown in this graph.(0.3)The function g(x) = -2x - 5.Compare the slopes.

Answer 1

We are given two functions to compare their slopes.

The following function f ( x ) is given in a form of a graph as a linear function. All linear function are expressed in a slope-intercept form as follows:

$f\text{ ( x ) = m}\cdot x\text{ + c}$

Where,

$\begin{gathered} m\text{ = slope} \\ c\text{ = y-intercept} \end{gathered}$

To determine the slope-intercept form of any linear function f ( x ) we need to determine the slope ( m ) and y-intercept ( c ). We can evaluate these two constants ( m and c ) using two pair of coordinates that lie on the staright line.

Considering the graph we have two points that lie on the line ( red-dots ). We will go ahead and write out these coordinates as follows:

$\begin{gathered} (x_1,y_1)\text{ = ( 0 , 3 )} \\ (x_2,y_2)\text{ = ( 1 , 1 )} \end{gathered}$

To determine the slope ( m ) of a linear function f ( x ) we will use the following formula:

$m\text{ = }\frac{y_2-y_1}{x_2-x_1}$

We will go ahead and plug in the respective coordinates in the above formulation of slope ( m ) as follows:

$\begin{gathered} m\text{ = }\frac{1\text{ - 3}}{1\text{ - 0}} \\ \\ m\text{ = }\frac{-2}{1} \\ \\ m\text{ = -2} \end{gathered}$

The y-intercept ( c ) of the line given as a graph can be directly evaluated by looking at the pair of coodinate where the straight line cuts or intersects the y-axis i.e:

$\begin{gathered} (\text{ 0 , 3 )} \\ \end{gathered}$

Hence,

$c\text{ = 3}$

Now we can write the slope-intercept form of the function f ( x ) as follows:

$f\text{ ( x ) = -2x + 3}$

Another linear function is given as g ( x ) as follows:

$g\text{ ( x ) = -2x - 5}$

Comparing this new function g ( x ) with the general slope-intercept form of the equation we can determine its slope as follows:

$g\text{ ( x ) = -2x - 5 }\equiv\text{ m}\cdot x\text{ + c}$

Where,

$\begin{gathered} m\text{ = -2} \\ c\text{ = -5} \end{gathered}$

We have slopes of both f ( x ) and g ( x ) as ( m ). Both values of ( m ) are the same! that is the slopes of the given graph and the function g ( x ) are exactly equal.

Select the option that is similar to above bold statement.