Investigation

• At a jazz club, the cost of an evening is based on a cover charge of $5 plus a beverage charge of $3 per drink.– Find a formula for t(x), the total cost for an evening out in which x

drinks are consumed– Glen the manager decides charge an extra dollar for cover into

the club. Explain how Glen’s decision would change t(x). Write a new formula (d(x)), using t(x), to describe the new situation that Glen created with his decision to raise the cover $1

– After Glen raised the cover, many of the patrons started to complain. Glen decides that he needs to create a positive experience. Glen’s assistant manager Paul comes up with the idea to still charge the same cover but give out two free drink tickets. How would this decision change t(x)? Write a new formula (h(x)), to describe the new situation.

Chapter 5 Section 1

PreCalc and Trig

Learning Targets

• I can explain a horizontal and vertical shift of a given function

• I can interpret a function in terms of shifts vs. real life scenarios.

Example

• Consider the table below the parent function f(x).

• How would the table change if the function was f(x) + 2?

• The output values are increasing by 2

x 1 2 3 4 5 6

f(x) -5 -3 0 3 7 12

t 1 2 3 4 5 6

f(t) -3 -1 2 5 9 14

Example

• Consider the table below the parent function f(x).

• How would the table change if the function was f(x) - 2?

• The output values are decreasing by 2

x 1 2 3 4 5 6

f(x) -5 -3 0 3 7 12

t 1 2 3 4 5 6

f(t) -7 -5 -2 1 5 10

Vertical Shifting

• Given the parent function f(x):– y = f(x) + k is the graph f(x) shifted vertically |k|

units.– If k is positive, the graph shifts up– If k is negative, the graph shifts down

• f(x) = x2 f(x) + 2 = x2 + 2

• f(x) = x2 f(x) – 2 = x2 – 2

Example

• Consider the table below the parent function f(x).

• How would the table change if the function was f(x + 2)?

• The input values are increasing by 2

x 1 2 3 4 5 6

f(x) -5 -3 0 3 7 12

t 1 2 3 4 5 6

f(t) 0 3 7 12 --- ---

Example

• Consider the table below the parent function f(x).

• How would the table change if the function was f(x – 2)?

• The input values are decreasing by 2

x 1 2 3 4 5 6

f(x) -5 -3 0 3 7 12

t 1 2 3 4 5 6

f(t) --- --- -5 -3 0 3

Horizontal Shifting

• Given the parent function f(x):– y = f(x + k) is the graph f(x) shifted horizontally |k|

units.– If k is positive, the graph shifts left– If k is negative, the graph shifts right

• f(x) = x2 f(x + 2) = (x + 2)2

• f(x) = x2 f(x – 2) = (x – 2)2

Example

• Explain the shift between f(x) and g(x).

• f(x) = 2-x + 1 g(x) = 2-(x+1) + 1

• Shift Left One

Example

• Explain the shift of the shift between h(x) and j(x)

• h(x) = x2 – 3 j(x) = x2 – 6x + 10

• Look at graphs

• h(x – 3) + 4 = j(x)