Question

The graphs of y = f(x) and y = g(x) are shown on the coordinate plane below. If g(x) = k* f(x), what is the value of k?

Answer 1

step 1
find the equation of f(x)
points
(2,1)  and (0,-3)
m=(y2-y1)/(x2-x1)----------> m=(-3-1)/(0-2)---------> m=2
y-y1=m(x-x1)-----> y+3=2*(x-0)------> y=2x-3
f(x)=2x-3

step 2
find the equation of g(x)
points 
(0,6)  and (2,-2)
m=(y2-y1)/(x2-x1)----------> m=(-2-6)/(2-0)---------> m=-4
y-y1=m(x-x1)-----> y-6=-4*(x-0)------> y=-4x+6
g(x)=-4x+6

step 3
find the value of k
If g(x) = k* f(x)
so
-4x+6=k*[2x-3]-------> -4x+6=2kx-3k
-4x=2kx--------> k=-2
6=-3k------------> k=-2

the answer is 
k=-2

Answer 2

The equation of line $f(x)$ and $g(x)$ shown in coordinate plane gives the value of constant $k$. Comparing both the equation of lines, we get the value of k is -2.

What is the equation of line?

The equation of the line is the way of representation of a line in the equation form.

The equitation of the lines represents the line by the set of points on which the line lies or passes through in the coordinate system.

Let a line passes through the point $(x_1,y_1)$ and $(x_2,y_2)$. Thus the equation of line can be given as,

$(y-y_1)=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)$

Given information-

The graph of $f(x)$ and $g(x)$ are given in the problem.

The equation given is,

$g(x)=k\times f(x)$

To find the value of $k$, the equation of  $f(x)$ and $g(x)$ has to be find out.

The line $y=f(x)$ lies on the points, (2,1) and (0,-3). Thus the equation of this line is,

$(y-(-3))=\dfrac{-3-1}{0-2}(x-0)\\y+3=\dfrac{-4}{-2}x\\y+3=2x\\y=2x-3$

It can be written in the form of $f(x)$ as,

$f(x)=2x-3$

The line $y=f(x)$ lies on the points, (0,6) and (2,-2). Thus the equation of this line is,

$(y-6)=\dfrac{-2-6}{2-0}(x-0)\\y-6=\dfrac{-8}{2}x\\y-6=-4x\\y=-4x+6$

It can be written in the form of $g(x)$ as,

$g(x)=-4x+6$

The equation given in the problem is,

$g(x)=k\times f(x)$

Put the values as,

$\begin{aligned}-4x+6&=k\times(2x-3)\\-2(2x-3)&=k\times(2x-3)\\\\\end$

Compare both the equation we get,

$k=-2$

Hence, the value of k is -2.

Learn more about the equation of line here;

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