Transforming Functions: Graph Transformations and Algebraic Changes, Exercises of Algebra

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Transformation of Functions

You should know the graph of the following basic functions:

f (x) = x^2

f (x) = x^3

f (x) =

x

f (x) =

x

Vertical Shift:

Given a function f (x), if c > 0 is a constant, then:

The graph of f (x) + c is the graph of f moved up c units.

The graph of f (x) − c is the graph of f moved down c units.

Example: Starting with f 1 (x) = x^2 , then f 2 (x) = f 1 (x) + 3 = x^2 + 3. The graph of f 2 (x) = x^2 + 3 is the graph of x^2 moved up by 3 units.

x^2 + 3

x^2

Example: Starting with f 1 (x) = x^2 , we can obtain f 2 (x) = f 1 (x) − 1 = x^2 − 1. The graph of x^2 − 1 is the graph of x^2 moved down by 1 unit.

x^2 − 1

x^2

Vertical stretch/compression

The graph of cf (x) is the graph of f stretched vertically (from the x−axis) by a factor of c if c > 1. To visualize a vertical stretch, imagine that you are pulling the graph of f away from the x−axis in both up and down directions. In a vertical stretch, the x−intercepts are unchanged.

Example: Starting with f 1 (x) =

x, we can obtain f 2 (x) = 2f 1 (x) = 2

x. The graph of 2

x is the graph of

x stretched by a factor of 2 vertically.

x

x

The graph of cf (x) is the graph of f compressed vertically by a factor of

c if c < 1 To visualize a vertical compression, imagine you push the graph of f toward the x−axis from both up and down directions. In a vertical compression, the x−intercepts are unchanged.

Example: Starting with f 1 (x) =

x, we can obtain f 2 (x) =

f 1 (x) =

x. The

graph of

x is the graph of

3 compressed by a factor of 3 vertically.

x

x

The graph of f (cx) is the graph of f strectched horizontally by a factor of

c if c < 1. To visualize a horizontal stretch, imagine that you pull the graph of the function away from the y−axis from both the left and the right hand side. In a horizontal stretch, the y−intercept is unchanged.

Example: Starting with f 1 (x) = x^3 , we can obtain f 2 (x) = f 1

x

x

The graph of

x

is the graph of x^3 stretched by a factor of 2 horizontally.

( 1 2

x

x^3

Again, pay attention to the fact that, in a horizontal stretch/compression that, if the constant c multiplied to x is greater than 1, the resulting graph is actually compressed by a factor of 1/c. On the other hand, if c is less than 1, the resulting graph is stretched by a factor of c.

Reflection

The graph of −f (x) is the graph of f reflected with respect to the x-axis. To draw a reflection with respect to the x−axis, draw the resulting graph by reflecting everything in the original graph using the x−axis as the mirror.

Example: Starting with f 1 (x) = x^2 , we can obtain f 2 (x) = −f 1 (x) = −x^2. The graph of −x^2 is the graph of x^2 reflected with respect to the x−axis:

x^2

−x^2

Example: Starting with f 1 (x) =

x, we can obtain f 2 (x) = −f 1 (x) = −

x. The graph of −

x is the graph of

x reflected with respect to the x−axis:

x

x

To draw the graph of a complicated function, start with a simple function whose graph is known. Algebraically change the known function one step at the time, by adding or subtracting a constant, multiply by a constant ...etc, to the function in question. The geometry must be dictated by the algebraic changes, not the other way around. You must first finish the algebra as to how the simple function changes to the complicated function, then do the geometry.

Example:

Graph f (x) =

− 2 x + 1 − 1

Ans:

We start with f 1 (x) =

x. We know the graph of f 1 (x)

Let f 2 (x) = f 1 (x + 1) =

x + 1. The graph of f 2 (x) can be obtained from the graph of f 1 (x) by a horizontal shift one unit to the left.

f 1 (x) =

x

f 2 (x) =

x + 1

f 3 (x) = f 2 (2x) =

2 x + 1. The graph of f 3 (x) can be obtained by horizontally compress f 2 (x) by a factor of 2.

f 3 (x) =

2 x + 1

f 2 (x) =

x + 1

f 4 (x) = f 3 (−x) =

2(−x) + 1 =

− 2 x + 1. The graph of f 4 (x) can be obtained by reflecting the graph of f 3 (x) with respect to the y−axis.

f 3 (x) =

f 4 (x) = 2 x + 1

− 2 x + 1

f 5 (x) = f 4 (x) − 1 =

− 2 x + 1 − 1. The graph of f 5 (x) is obtained by vertically shift f 4 (x) one unit down.

f 4 (x) =

− 2 x + 1

f 5 (x) =

− 2 x + 1 − 1

The graph of f is obtained from the graph of

x by moving left one unit, followed by horizontal compression by a factored of 2, reflected with respect to the y axis, then moved down one unit.

g 4 (x) = g 3

x −

x −

− 2 x + 1. The graph of g 4 (x) is

obtained by moving g 3 (x) half unit to the right.

g 4 (x) =

− 2 x + 1

g 3 (x) =

− 2 x

Notice in this step that we added −

to the argument of g 3 , not 1. The reason

is because if we added 1 to the argument of g 3 , the distributive property would give us

− 2 x − 2, which is not what we want. You must always actually work out the algebra of how each subsequent function is obtained from the previous function.

g 5 (x) = g 4 (x) − 1 =

− 2 x + 1 − 1. The graph of g 5 (x) is obtained by moving the graph of g 4 (x) 1 unit down.

g 4 (x) =

− 2 x + 1

g 5 (x) =

− 2 x + 1 − 1

The graph of f can be obtained from the graph of

x by first horizontally

compress by a factor of 2, reflected with respect to y, moved to the right by

unit, then moved down 1 unit.

Example: Draw the graph of −

3 x − 1

Ans: We start with f 1 (x) =

x

f 2 (x) = f 1 (x − 1) =

x − 1

. The graph of g 2 (x) can be obtained by horizontally

shift f 1 (x) 1 unit to the right.

f 1 (x) =

x

f 2 (x) =

x − 1

f 4 (x) = 2f 3 (x) = 2

3 x − 1

3 x − 1

. The graph of f 4 (x) can be obtained by

vertically stretch f 3 (x) by a factor of 2.

f 4 (x) =

3 x − 1

f 3 (x) =

3 x − 1

f 5 (x) = −f 4 (x) = −

3 x − 1

. The graph of f 4 (x) can be obtained by reflecting

f 4 (x) with respect to the x−axis.

f 4 (x) =

3 x − 1 f 5 (x) = −

3 x − 1