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

Consider this option:
1. according to the attached graph the equation for line f is f(x)=2x-3 (it is easy to determine it using coordinates of points (0;-3) and (2;1));
2. according to the graph the equation for line g is g(x)=-4x+6 (using coordinates of points (0;6) and (2;-2)).
3. according to the condition g(x)=k*f(x), k=g(x)/f(x)=(-4x+6)/(2x+3)=-2.

answer: -2

Answer 2

Value of k = -2

Explanation:

Using slope intercept form:

The equation of line is given by:

y = mx+b

where, m is the slope and b is the y-intercept.

As per the statement:

The graphs of y = f(x) and y = g(x) are shown on the coordinate plane.

First find f(x):

Consider two points on y=f(x) are:

(0,-3) and (2, 1)

Formula for slope:

`\text{Slope(m)} = (y_2-y_1)/(x_2-x_1)`

then;

`m = (1-(-3))/(2-0)=(4)/(2) = 2`

then;

`y = 2x+b`

Substitute the point (0, -3) to solve for b:

`-3= 2(0)+b`

⇒
`-3=b`

∴
`y=f(x)=2x-3`

Similarly for g(x):

Consider two points on y = g(x) are:

(0, 6) and (2, -2)

then;

`\text{Slope(m)} = (y_2-y_1)/(x_2-x_1)`

then;

`m = (-2-6)/(2-0)=(-8)/(2) =-4`

then;

`y = -4x+b`

Substitute the points (0, 6) to solve for b:

`6= -4(0)+b`

⇒
`6=b`

then we get an equation:

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

It is given that: If g(x) = k* f(x)

Solve for k:

`-4x+6 = k(2x-3)`

⇒
`-4x+6 = 2kx-3k`

on comparing both sides we have;

`-2kx = -4x`

⇒
`-2k = -4`

Divide both sides by -2 we have;

`k = 2`

or

-3k = 6

⇒k = -2

Therefore, the value of k is, -2