Exercise 41 Page 18

Practice makes perfect

a In order to find the vertical translation of the graph of

f

to the graph of

g,

we need to use the vertical translation rule two times. First, we shift the graph of

f

b

units.

h (x) = f (x) + k

$f (x) = m x + b$ , $k = - b$

h (x) = m x + b - b

Subtract term

h (x) = m x

Now we have to shift the graph of

h

c

units.

g (x) = h (x) + k

$f (h) = m x$ , $k = c$

h (x) = m x + c

If we combine both translations, we get the expression for the vertical translation of

f

g .

g (x) = f (x) + c - b

b Again, we want to translate the graph of

f

to the graph of

g .

We know that the factor to do so is

c - b

. However, this time we want to translate horizontally. Let's try it out with the factor of

c - b .

g (x) = f (x + h)

$f (x) = m x + b$ , $h = b - c$

g (x) = m (x + c - b) + b

Distribute $m$

g (x) = m x + m c - m b + b

We see that whatever translating factor we choose will be multiplied by the slope

m .

This means that we have to divide our translating factor by

m

to compensate.

\frac{c - b}{m}

Let's see if it works now.

g (x) = f (x + h)

$f (x) = m x + b$ , $h = \frac{c - b}{m}$

g (x) = m (x + \frac{c - b}{m}) + b

Distribute $m$

g (x) = m x + \frac{m ( c - b )}{m}) + b

$\frac{a}{b} = \frac{a / m}{b / m}$

g (x) = m x + c - b + b

Add terms

g (x) = m x + c

Since we got the right result,

g (x) = f (x + \frac{c - b}{m})

is the expression that horizontally translates

f

g .

Communicate Your Answer p.11

Monitoring Progress p.13-15

Exercises p.16-18

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