Download 1-3 Transforming Linear functions

CHAPTER 1
1-3 TRANSFORMING LINEAR FUNCTIONS
WARM UP
Instructions: Name the parent function of the
following problems:
1.y = 3𝑥 2 + 15
2. 𝑦 =
1
𝑥
1
+
4
3. y = 3𝑥
1
+
2
4. 𝑦 = 2 3𝑥
WARM UP ANSWER
Instructions: Name the parent function of the
following problems:
1.y = 3𝑥 2 + 15
Answer: Quadratic Function
1
𝑥
2. 𝑦 = +
1
4
Answer: Rational Function
3. y = 3𝑥 +
1
2
Answer: Linear function
4. 𝑦 = 2 3𝑥
Answer; Square root function
OBJECTIVES
• The student will be able to:
• Transform linear functions
• Solve problems involving linear transformations
TRANSFORMING LINEAR FUNCTIONS
• What is a transformation?
• Answer: A transformation is a change in the position, size,
or shape of a figure or graph.
• What is a Linear function?
• Answer: is a function, meaning we have an input and an
output, that can be written in the form 𝑓 𝑥 = 𝑚𝑥 + 𝑏. Its
graph is a line.
• If we transforming linear functions , we can say we are
changing the linear function either the way it looks in the
graph or the equation.
TRANSFORMING LINEAR FUNCTIONS
• There are four ways we can transform the linear
function by :
• Just remember the x changes
TRANSFORMING LINEAR FUNCTIONS
Just remember y changes
TRANSFORMING LINEAR FUNCTIONS
Just remember y is the mirror so the one that changes is
the x
TRANSFORMING LINEAR FUNCTIONS
Just remember x is the mirror so the one that changes is
the y
EXAMPLE 1
• Let g(x) be the indicated transformation of
f(x).Write the rule for g(x).
• 𝑓 𝑥 = 3𝑥 + 2; g(x) is a horizontal shift 3 units to the
right.
• Solution:
• 𝑔 𝑥 = 𝑓 𝑥 − 3 subtract 3 from the input
• 𝑔 𝑥 = 3 𝑥 − 3 + 2 evaluate f at x-3
• 𝑔 𝑥 = 3𝑥 − 9 + 2 Simplify
• 𝑔 𝑥 = 3𝑥 − 7
EXAMPLE 2
• Let g(x) be the indicated transformation of
f(x).Write the rule for g(x).
• 𝑓 𝑥 = 𝑥 + 2; g(x) is reflected about the y-axis.
• Solution:
• 𝑔 𝑥 = 𝑓(−𝑥) change the input of f
• 𝑔 𝑥 = −𝑥 + 2 Simplify
• 𝑔 𝑥 = −𝑥 + 2
STUDENT PRACTICE
EXAMPLE 3
• Let g(x) be the indicated transformation of
f(x).Write the rule for g(x).
• 𝑓 𝑥 = 6𝑥 + 2; g(x) is a vertical shift (vertical
translation) 3 units down.
STUDENT PRACTICE
EXAMPLE 4
• Let g(x) be the indicated transformation of
f(x).Write the rule for g(x).
• 𝑓 𝑥 = 6𝑥 + 2; g(x) is a reflection across the x-axis.
LETS COMBINE TRANSFORMATIONS
EXAMPLE 5
• Let g(x) be the indicated transformation of
f(x).Write the rule for g(x).
• 𝑓 𝑥 = 2𝑥 − 6; g(x) is a vertical shift (vertical
translation) 3 units down followed by a reflection
across the x-axis
• .Solution:
• First lets take care of the vertical translation
• 𝑔 𝑥 =𝑓 𝑥 −3
• 𝑔 𝑥 = 2𝑥 − 6 − 3 substitute
• 𝑔 𝑥 = 2𝑥 − 9 simplify
EXAMPLE 5 CONTINUE
• Then we continue with the reflection across the xaxis
• 𝑔 𝑥 = −𝑓 𝑥
• 𝑔 𝑥 = − 2𝑥 − 9
• 𝑔 𝑥 = −2𝑥 + 9
STRETCHES AND COMPRESSION
• Stretches and compressions change the slope of a
linear function. If the line becomes steeper, the
function has been stretched vertically or
compressed
• horizontally. If the line becomes flatter, the function
has been compressed vertically or stretched
horizontally.
STRETCHES AND COMPRESSIONS
EXAMPLE 6
• Let g(x) be a vertical compression of f(x) = 3x + 2 by
a factor of 4 . Write the rule for g(x) and graph the
function.
• Solution:
• Vertically compressing f(x) by a factor of replaces
each f(x) with a · f(x) where a = 4 .
• 𝑔 𝑥 =𝑎∗𝑓 𝑥 =4∗𝑓 𝑥
• 𝑔 𝑥 = 4 ∗ (3𝑥 + 2) substitute
• 𝑔 𝑥 = 12𝑥 + 8 simplify
STUDENT PRACTICE
EXAMPLE 7
• Let g(x) be a horizontal compression of f(x) = 5x - 2
by a factor of 1/3 . Write the rule for g(x) and
graph the function.
NOW LETS PUT EVERYTHING
TOGETHER
Example 8:
Let g(x) be a horizontal compression of f(x) = 6x - 5by
a factor of 1/3 followed by a vertical translation 4
units up . Lets h(x) be the horizontal compression and
g(x) the vertical translation. Write the rule for g(x) and
graph the function.
1
𝑓 𝑥 =ℎ 𝑥
𝑏
1
𝑓 𝑥 =ℎ
𝑥 = ℎ 3𝑥
1
3
ℎ 𝑥 = 6 3𝑥 − 5 = 18𝑥 − 5
EXAMPLE 8 CONTINUE
•
•
•
•
Now lets take care of the translation
𝑔 𝑥 =ℎ 𝑥 +4
𝑔 𝑥 = 18𝑥 − 5 + 4 substitute
𝑔 𝑥 = 18𝑥 − 1 simplify
STUDENT PRACTICE
• Do all worksheet
•
HOMEWORK
• Page 28 from book
• problems 2 to 6 and 12 to14.
CLOSURE
• Today we talked about transforming linear functions
through translating and reflecting .
• Tomorrow we are going to see scatter plots and the
best fit line.
HAVE A GREAT DAY!!!