Answer:
A linear function can be written as:
y = a*x + b
Where a is the slope and b is the y-intercept.
If this line passes through the points (x₁, y₁) and (x₂, y₂), then the slope is:
a = (y₂ - y₁)/(x₂ - x₁)
We know that line f(x) passes through points (5, 0) and (10, 10)
Then the slope of f(x) is:
a = (10 - 0)/(10 - 5) = 10/5 = 2
then:
f(x) = 2*x + b
again, knowing that f(10) = 10 (because this line passes through point (10, 10) )
10 = 2*10 + b
10 = 20 + b
10 - 20 = b = -10
Then the equation of this line is:
f(x) = 2*x - 10
Now for g(x) we know that it passes through (-3, 0) and (2, 10)
then the slope is:
a = (10 - 0)/(2 - (-3)) = 10/5 = 2
g(x) = 2*x + b
And knowing that g(2) = 10 (because this line passes through point (2, 10) )
10 = 2*2 + b
10 = 4 + b
10 - 4 = b
6 = b
then:
g(x) = 2*x + 6
Then the equations are:
f(x) = 2*x - 10
g(x) = 2*x + 6
A) One transformation may be a vertical translation, and the other can be a horizontal translation.
This is a vertical translation, which is defined as:
For a function f(x), a vertical translation of k units is written as:
g(x) = f(x) + k
If k is positive, the translation is upwards
if k is negative, the translation is downwards.
And an horizontal translation of k units is written as:
g(x) = f(x - k)
if k is positive, the translation is to the right
if k is negative, the translation is to the left.
B)
For the vertical translation we have:
g(x) = f(x) + k
2*x + 6 = 2*x - 10 + k
6 = -10 + k
6 + 10 = k
16 = k
For the horizontal translation we have:
g(x) = f(x - k)
2*x + 6 = 2*(x - k) - 10
2*x + 6 = 2*x - 2*k - 10
6 = -2*k - 10
6 + 10 = -2*k
16 = -2*k
16/-2 = k
-8 = k
C)
The transformations are:
g(x) = f(x) + 16
or
g(x) = f(x - (-8))
