Question

The graph of the linear function \[f\] has intercept at $\left( {a,0} \right)$ and $\left( {0,b} \right)$ in the $xy$ plane. If \[a + b = 0\] and \[a \ne b\], which of the following is true about the slope of the graph of \[f\] ?
1) It is positive.
2) It is negative
3) It equals zero
4) It is undefined

Answer 1

Hint: The first step will be to form the equation of the linear function using the two points that are given to us and to determine the slope from the equation in terms of $a$ and $b$. The relation \[a + b = 0\] can be then used to find the value of the slope of the function.

Complete step by step answer:

It is known that the equation for the linear function can be determined by two points. As we know that an equation of the linear function passing through points say \[\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[\left( {{x_2},{\text{ }}{y_2}} \right)\] is given by
\[x - {x_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {y - {y_1}} \right)\]
We are given the question that the linear function \[f\] passes through the two points $\left( {a,0} \right)$ and $\left( {0,b} \right)$.
Therefore equation of the linear function can be written as
\[x - a = \dfrac{{b - 0}}{{0 - a}}\left( {y - 0} \right)\]
On rearranging the equation, we get
$
  x - a = - \dfrac{b}{a}y \\
  ax - {a^2} = - by \\
$
On writing the equation in the standard form of the linear function, \[y = mx + c\], we get
\[y = - \dfrac{a}{b}x + \dfrac{{{a^2}}}{b}\]
Comparing the equation of the linear function with the standard equation of the linear function that is \[y = mx + c\] where, $m$ is the slope of the function, we can say that the slope of the given linear function is
$m = - \dfrac{a}{b}$
Here we have to comment on the nature of the slope.
We are given the equation \[a + b = 0\]. On rearranging the equation we get \[a = - b\].
We can solve for the slope of the linear function by substituting the value \[ - b\] for \[a\] in the equation of the slope $m = - \dfrac{a}{b}$.
$
  m = - \dfrac{{\left( { - b} \right)}}{b} \\
  m = 1 \\
 $
Thus, we can say that the slope of the given linear function is 1 which is positive.
Thus the correct answer is option A. It is positive.

Note: Another way to solve the question is by plotting graphs for a pair of \[a\] and \[b\] that satisfies the equation \[a + b = 0\]. The slope of the graph then can be evaluated from the graph. We have observed that the slope of the function is a fixed value that is 1.