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= 3 𝑥 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= 3 𝑥 2 +15 Answer: Quadratic Function 2. 𝑦= 1 𝑥 + 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 x-axis 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. 𝑓 𝑥 =ℎ 1 𝑏 𝑥 𝑓 𝑥 =ℎ 1 1 3 𝑥 =ℎ 3𝑥 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!!!