USE TRANSFORMATIONS TO GRAPH THE FUCTION FROM GIVEN GRAPH

Considering the given function in the form

y = a (x - h) + k

Based on the signs, we decide what transformation can be made.

  • If h > 0, then move the graph horizontally h units right.
  • If h < 0, then Move the graph horizontally h units left.
  • If k > 0, then move the graph k units up.
  • If k < 0, then move the graph k units down.
  • If a > 1, vertical stretch of a units.
  • If 0 < a < 1, vertical shrink of a units.

Reflection :

  • If f(x) = f(-x), then reflection across y-axis.
  • If f(x) = -f(x), then reflection across x-axis.

Horizontal stretch or shrink :

If f(x) = f(ax),

then horizontal shrink of a units.

If f(x) = f(ax)

  • a > 0 then horizontal shrink of a units.
  • 0 < a < 1, horizontal stretch of of a units.


The graph of y = f(x) is given below. Sketch the graph of each of the following functions.

using-transformation-graph-q1

Problem 1 :

y = f(x + 2)

Solution :

y = f(x + 2)

y = f(x - (-2))

Here h = -2

Move the graph horizontally 2 units left.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A'(-5-2, -2) ==> A'(-7, -2)

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

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

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

E'(8-2, 0) ==> E'(6, 0)

using-transformation-graph-q1answer.png

Problem 2 :

y = f(x) - 3

Solution :

y = f(x) - 3

Comparing with y = a f(x-h) + k

Here k = -3

Move the graph vertically 3 units down.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A'(-5, -2-3) ==> A'(-5, -5)

B'(-2, -2-3) ==> B'(-2, -5)

C'(0, 0-3) ==> C'(0, -3)

D'(4, 4-3) ==> D'(4, 1)

E'(8, 0-3) ==> E'(8, -3)

using-transformation-graph-q2.png

Problem 3 :

y = f(x - 2) - 1

Solution :

y = f(x - 2) - 1

Comparing with y = a f(x-h) + k

Here h = 2 and k = -1

Move the graph horizontally 2 units right and 1 unit down.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A'(-5+2, -2-1) ==> A'(-3, -3)

B'(-2+2, -2-1) ==> B'(0, -3)

C'(0+2, 0-1) ==> C'(2, -1)

D'(4+2, 4-1) ==> D'(6, 3)

E'(8+2, 0-1) ==> E'(10, -1)

using-transformation-graph-q3.png

Problem 4 :

y = f(-x)

Solution :

y = f(-x)

Here x is changed as -x. So, reflection across y axis.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A(-(-5), -2) ==> A'(5, -2)

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

C(0, 0) ==> C'(0, 0)

D(-4, 4) ==> D'(-4, 4)

E(-8, 0) ==> E'(-8, 0)

using-transformation-graph-q4.png

Problem 5 :

y = -f(x)

Solution :

y = -f(x)

Here y is changed as -y. So, reflection across x axis.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A(-5, -(-2)) ==> A'(-5, 2)

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

C(0, 0) ==> C'(0, 0)

D(4, -4) ==> D'(4, -4)

E(8, 0) ==> E'(8, 0)

using-transformation-graph-q5.png

Problem 6 :

y = 2f(x)

Solution :

y = 2f(x)

Vertical stretch of 2 units.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A(-5, 2(-2)) ==> A'(-5, -4)

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

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

D(4, 2(4)) ==> D'(4, 8)

E(8, 2(0)) ==> E'(8, 0)

using-transformation-graph-q6.png

Problem 7 :

y = (1/2)f(x)

Solution :

y = (1/2)f(x)

Vertical shrink of 1/2 units.

A(-5, -2)

B(-2, -2)

C(0, 0)

D(4, 4)

E(8, 0)

A(-5, -2/2) ==> A'(-5, -1)

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

C(0, 0/2) ==> C'(0, 0)

D(4, 4/2) ==> D'(4, 2)

E(8, 0/2) ==> E'(8, 0)

using-transformation-graph-q7.png

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