GRAPHING EXPONENTIAL FUNCTIONS WITH TRANSFORMATIONS

Given the graphed parent function y = 2^x, perform the following transformations.

Problem 1 :

y = 2^{x - 2}

Solution :

y = 2^{x - 2}

Horizontal translation should be done, move the graph 2 units to the right.

From the graph, listing out some of the points and performing transformation.

A(0, 1) ==> A' (0 + 2, 1) ==> A' (2, 1)

B(1, 2) ==> B' (1 + 2, 2) ==> B' (3, 2)

C(2, 4) ==> C' (2 + 2, 4) ==> C' (4, 4)

D(3, 8) ==> D' (3 + 2, 8) ==> D' (5, 8)

Problem 2 :

y = 2^x - 2  

Solution :

y = 2^x - 2 

Vertical translation should be done, move the graph 2 units down.

From the graph, listing out some of the points and performing transformation.

A(0, 1) ==> A' (0, 1 - 2) ==> A' (0, -1)

B(1, 2) ==> B' (1, 2 - 2) ==> B' (1, 0)

C(2, 4) ==> C' (2, 4 - 2) ==> C' (2, 2)

D(3, 8) ==> D' (3, 8 - 2) ==> D' (3, 6)

Problem 3 :

y = (-1)2^x

Solution :

y = (-1)2^x 

Reflection is done, y is changed as -y. So, it is reflection across x-axis.

From the graph, listing out some of the points and performing transformation.

A(0, 1) ==> A' (0, -1)

B(1, 2) ==> B' (1, - 2)

C(2, 4) ==> C' (2, -4)

D(3, 8) ==> D' (3, -8)

Problem 4 :

y = 2^x+2

Solution :

y = 2^x+2 

Horizontally the parent function is moved towards left 2 units.

From the graph, listing out some of the points and performing transformation.

A(0, 1) ==> A' (0 - 2, 1) ==> A'(-2, 1)

B(1, 2) ==> B' (1 - 2,  2)==> B' (-1,  2)

C(2, 4) ==> C' (2 - 2, 4)==> C' (0, 4)

D(3, 8) ==> D' (3 - 2, 8)==> D' (1, 8)

Problem 5 :

y = 2^x + 3  

Solution :

y = 2^x + 3

Vertical translation should be done, move the graph 3 units up.

From the graph, listing out some of the points and performing transformation.

A(0, 1) ==> A' (0, 1 + 3) ==> A' (0, 4)

B(1, 2) ==> B' (1, 2 + 3) ==> B' (1, 5)

C(2, 4) ==> C' (2, 4 + 3) ==> C' (2, 7)

D(3, 8) ==> D' (3, 8 + 3) ==> D' (3, 11)

Problem 6 :

y = (-2)2^x

Solution :

y = (-2)2^x

Reflection is done. y is changed as -y. So, reflection across x axis. The scale factor is 2, 2 > 1. It must be a vertical stretch.

From the graph, listing out some of the points and performing transformation.

A(0, 1) ==> A' (0, -2(1)) ==> A' (0, -2)

B(1, 2) ==> B' (1, -2(2)) ==> B' (1, -4)

C(2, 4) ==> C' (2, -2(4)) ==> C' (2, -8)

D(3, 8) ==> D' (3, -2(8)) ==> D' (3, -16)