GRAPHING FUNCTIONS USING VERTICAL AND HORIZONTAL SHIFTS

Example 1 :

Sketch the graph of

g(x) = |x + 2|

Solution :

The basic function for the given function is

f(x) = |x|

or

y = |x|

To find the points on the basic function, take some random values for x and find the corresponing values for y.

When x = -2,

y = |-2| = 2

When x = -1,

y = |-1| = 1

When x = 0,

y = |0| = 0

When x = 1,

y = |1| = 1

When x = 2,

y = |2| = 2

Points on the graph of the basic function :

(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)

Graph of the basic function f(x) = |x| :

g(x) = |x + 2|

Since f(x) = |x|, we have

g(x) = f(x + 2)

Here, 2 is added to x. So, there is an horizontal shift of 2 units to the left.

We have to shif the graph of the basic function 2 units to the left to get the graph of g(x) = |x + 2|.

Graph of g(x) = |x + 2| :

Example 2 :

Sketch the graph of

h(x) = √x + 2

Solution :

The basic function for the given function is

f(x) = √x

or

y = √x

Here, √x is defined for x ≥ 0.

To find the points on the basic function, take some random values for x and find the corresponing values for y.

When x = 0,

y = √0 = 0

When x = 1,

y = √1 = 1

When x = 4,

y = √4 = 2

Points on the graph of the basic function :

(0, 0), (1, 1), (4, 2)

Graph of the basic function f(x) = √x :

h(x) = √x + 2

Since f(x) = √x, we have

h(x) = f(x) + 2

Here, 2 is added to f. So, there is a vertical shift of 2 units up.

We have to shif the graph of the basic function 2 units up to get the graph of h(x) = √x + 2.

Graph of h(x) = √x + 2 :

Example 3 :

Sketch the graph of

j(x) = x² + 2

Solution :

The basic function for the given function is

f(x) = x²

or

y = x²

To find the points on the basic function, take some random values for x and find the corresponing values for y.

When x = -2,

y = (-2)² = 4

When x = -1,

y = (-1)² = 1

When x = 0,

y = 0² = 0

When x = 1,

y = (-1)² = 1

When x = 2,

y = (-2)² = 4

Points on the graph of the basic function :

(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)

Graph of the basic function f(x) = x² :

j(x) = x² + 2

Since f(x) = x², we have

j(x) = f(x) + 2

Here, 2 is added to j. So, there is a vertical shift of 2 units up.

We have to shif the graph of the basic function 2 units up to get the graph of j(x) = x² + 2.

Graph of j(x) = x² + 2 :

Example 4 :

Sketch the graph of

k(x) = |x - 2| - 1

Solution :

The basic function for the given function is

f(x) = |x|

or

y = |x|

To find the points on the basic function, take some random values for x and find the corresponing values for y.

When x = -2,

y = |-2| = 2

When x = -1,

y = |-1| = 1

When x = 0,

y = |0| = 0

When x = 1,

y = |1| = 1

When x = 2,

y = |2| = 2

Points on the graph of the basic function :

(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)

Graph of the basic function f(x) = |x| :

k(x) = |x - 2| - 1

Since f(x) = |x|, we have

g(x) = f(x - 2) - 1

Here, 2 is added to x. So, there is an horizontal shift of 2 units to the left.

1 is subtracted from f. So, there is a vertical shift of 1 unit down.

We have to shif the graph of the basic function 2 units to the right and 1 unit up to get the graph of 

k(x) = |x - 2| - 1

Graph of k(x) = |x - 2| - 1 :

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