Linear Functions: Vertical and Horizontal Translations

Linear Functions:  

Question:

Graph the function f (x) = – 2/3 x + 5 by plotting points.

Solution:

Evaluate the function at each input value and use the output value to identify ordered pairs.

For x=0, f (0) = -2/3(0) +5 =5……..(0,5)

For x=3, f (3) = -2/3(3) +5 =5……..(3,3)

For x=6, f (6) = -2/3(6) +5 =5……..(6,1)

Plot the ordered pairs and draw a line through the points that represent the graph of the function f(x) = -2/3 x + 5

Question:

Find the slope and Y-intercept of the function g (x) = 5x-3.

Solution:

The slope intercept form is given by function f (x) = mx + b where m is the slope and b is the y- y-intercept.

From the function g (x) = 5x-3

Slope, m = 5 and Y-intercept, B= -3

Question:

Find the linear function that models the data.

Solution:

The slope-intercept form: y = mx + b 

The constant rate of change is – 6. So, m=20 

y= 20 x + b

Substitute any ordered pair from the table.

-4 = 20 x 2 + b

-44 = b

So, the linear function is f (x) = 20x-44

Vertical Translation of Linear Functions:

Vertical translation:

For a given g (x) = f (x) + k, the graph of the function g is the function f that translates k units vertically.

k > 0: shifts |k| Units up

k < 0: shifts |k| Units down

Example: Compare the graph of the function: p (x) = x/5 and q (x) =x/5 -2

When compared, the graph q (x) = x/5 -2 has moved 2 units down to the graph p (x) = x/5.

Example: Compare the graph of the function: f (x) = -x and g (x) =-x + 1 

When compared, the graph g (x) =-x + 1 has moved 1 unit up to the graph f (x) = -x.

Example: How does the graph of the function f (x) = -3x translated to obtain the graph of g (x) = -3x + 2?

Solution:

g(x) = -3x + 2

g (x) = f (x) + 2

k =2; shifts |2| Units Down.

So, the graph of the function g is the function f that translates 2 units up vertically

Horizontal Translation of Linear Functions:  

Horizontal translation:

For a given,  g (x) = f(x –k), the graph of the function g is the function f that translates k units horizontally.

k > 0: shifts |k| Units right

k < 0: shifts |k| Units left

Example: Compare the graph of the function: f (x) =3x  and g (x) = 3 (x+1).

When compared, the graph g (x) = 3 (x+1) has moved 1 unit left to the graph f (x) =3x

Example: Compare the graph of the function:  f(x) = 2x-1 and g (x) =2 (x-3) -1.

When compared, the graph g (x) =2 (x-3) -1 has moved 3 units right to the graph f(x) = 2x-1.

Example: How the graph of the function f (x) = 3x-5 translated to obtain the graph of g (x) =3(x+7)-5?

Solution: g (x) =3(x+7) -5

 g (x) = f (x + 7) 

k = -7; shifts | -7 | units left

So, the graph of the function g is the function f that translates 7 units left horizontally.

Vertical Stretches and Compressions of Linear Functions: 

For any function f (x), multiplying the output of a linear function f by k scales which is (x) = kf (x), its graph is vertically stretched or compressed.

• If f | k| >1, the transformed graph is a vertical stretch,
• If 0 < | k | <1, the transformed graph is a vertical compression.

Example:

Consider the values of the functions f(x) = x +2 and g (x) = 4(x+2).

The graph of the functions f (x) = x+2 and g (x) = 4 (x + 2).

Here, the graph of g is a vertical stretch of the graph of f, by a scale factor of 4.

The slope and y-intercept are scaled by the same factor.

Question:

How does the graph of g (x) = 0.75 (x+2) compare with the graph of f (x) = x+2??

Solution:

Given g (x) = 0.75 (x+2) = 0.75 f (x)

When compared to g (x) = k f (x), where k = 0.75, where 0 < | k | <1

So, the graph of g is a vertical compression of the graph of f.

Horizontal Stretches and Compressions of Linear Functions: 

For any function f(x), multiplying the input of a linear function f by k scales which is g (x) = f k (x), its graph is horizontally stretched or compressed.

• If| k | > 1, the transformed graph is a horizontal compression.
• If 0 < | k | < 1, the transformed graph is a horizontal stretch.

Consider the values of the functions f (x) = x+ 2 and g (x) = 3x + 2.

The graph of the functions f (x) = x+ 2 and g (x) = 3x + 2.

Here, the graph of g is a horizontal compression of the graph of f, by a scale factor of 3.

The slope is changed by the same factor, but the y-intercept is unchanged.

Question:

How does the graph of g (x) = 0.3 x-1 compare with the graph of f (x) = x-1??

Solution:

Given  g (x) = 0.3 x-1= f (0.3)x

When compared to ,g (x) = f(k x),k = 0.3, where 0< |k| <1.

So, the graph of g is a horizontal stretch of the graph of f.

Exercise

• Q1: Describe how the graph of g (x) = 15 x- 5 compares with the graph of f (x) = 3x-1
• Q2: Describe how the graph of g (x) = 15 + 2 compares with the graph of f (x) = 5x + 2.
• Q3: Describe how the graph compares g (x) = -3x-1 with the graph of f (x) = – 3 (x).
• Q4: Given f (x) = 3x-1, describe how the value of k affects the slope and y-intercept of the graph of g (x) = 3(1/4) x-1 compared to the graph of f.
• Q5: Given f (x) = 2x+ 7, describe how the value of k affects the slope and y-intercept of the graph of g (x) = (2x + 7) -5 compared to the graph of f.

Concept Map:

What We Have Learned:

• Analyze vertical translation of linear functions.
• Analyze horizontal translation of linear functions.
• Analyze vertical stretches and compressions of linear functions.
• Analyze horizontal stretches and compressions of linear functions.