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Lesson #26 – Intro to Transformations on Functions

A2.A.46 Perform transformations with functions and relations: f(x+a), f(x)+a, f(–x), -f(x), af(x)

You will recall that we learned about parent functions last unit. Now we will learn about their

children or transformations. Let’s start with the parent function xxf )( .

xxf )( . Graph f(x) as Y1 on your calculator and the transformation as Y2.

Explain what the

notation indicates

Transformed

Function

Rough Sketch Domain and

Range

Name of the

Transformation

f(x)

************

*************

-f(x) y x

f(-x) y x

af(x)

Let a=2.

2y x

f(x+a)

5y x

f(x)+a 5y x

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Next we will perform the same transformations with the quadratic parent function. 2( )j x x Graph f(x) as Y1 on your calculator and the transformation as Y2.

Explain what the

notation indicates

Transformed

Function

Rough Sketch Domain and

Range

Name of the

Transformation

j(x)

***********

***********

-j(x)

j(-x)

aj(x)

Let a = 3.

j(x+a)

Let a = -4

j(x)+a

Let a = -2

Why was the second transformation the same as the original parent function?

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Next we will perform the same transformations with the absolute value parent function.

xxm )( . See if you can complete the table without the calculator.

Explain what each

notation indicates

Transformed

Function

Rough Sketch Domain and

Range

Transformation

Name

m(x)

***********

***********

-m(x)

m(-x)

a m(x)

a = .5

m(x+a)

a = 5

m(x)+a

a = -3

1)

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2)

3)

4)

Function Notation Transformation -f(x) Reflection in the x-axis

f(-x) Reflection in the y-axis

af(x) Vertical stretch of a

f(x+a) Translation left (+a) or right (-a)

f(x)+a Translation up (+a) or down (-a)

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Lesson #27 – Multiple Transformations A2.A.46 Perform transformations with functions and relations:

f(x+a), f(x)+a, -f(x), f(–x), af(x)

A2.A.51 Determine the domain and range of a function from its graph

A2.A.39 Determine the domain and range of a function from its

equation

Review

Function Notation Transformation -f(x) Reflection in the x-axis

f(-x) Reflection in the y-axis

af(x) Vertical stretch of a

f(x+a) Translation left (+a) or right (-a)

f(x)+a Translation up (+a) or down (-a)

The following activity will help you start to think about multiple transformations.

Identify the parent function for each transformation.

Next draw arrows to each transformation that has happened to the parent function and

identify them.

1. ( ) 3 7g x x

2. 3 7y x

3. ( ) 2h x x

4. 9

3

5)(

xxr

5.

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6.

7.

8.

In addition, you can also perform one of the transformations on a function that is not a parent

function. With these more complex functions it is important to follow the order of operations

when simplifying the parent function. Graph it on your calculator to check.

9.

10.

11.

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Last unit we also learned the domain and range for each of the parent functions. These values

can change after transformations, but the same principles apply. Since you are just getting

used to these concepts, it is often best to look at the graph of a transformed equation to find

its domain and range EXCEPT when you are working with rational functions. You should use the

rules for restricted domains on these.

12.

13.

14.

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The equation of a circle in

center-radius form

With Center (h,k): 2 2 2( ) ( )x h y k r

With Center (0,0): 222 )0()0( ryx

2 2 2x y r

Lesson #28 - Equations of Circles: 2 2and y added togetherx A2.A.48 Write the equation of a circle, given its center and a point

on the circle

A2.A.49 Write the equation of a circle from its graph

Distance formula: 2

12

2

12 )()( yyxxd

A circle is all the points, (__,__) a certain distance from the __________, (__, __). This

distance is called the __________.

Plug these values into the distance formula. Use (h,k) as the

first point and (x,y) as the second point.

Square both sides. This is your basic circle equation.

1. Write an equation describing all of the points exactly 6 units away from the origin.

2. Write the equation of a circle with radius 5 centered at (2,-5).

3. Graph 43422

yx

Domain:

Range:

Are circles functions? Why or why not?

4. Write the equation of a circle centered at (-1,-3) with radius 7.

A) B) C) D) E) F) G) H) I) J) K) L) M) N) O)

P) Q) R) S) T) U) V) W) X) Y) Z) AA) BB) CC) DD)

EE) FF) GG) HH) II) JJ) KK) LL) MM) NN) OO) PP) QQ) RR) SS)

TT) UU) VV) WW) XX) YY) ZZ) AAA) BBB) CCC) DDD) EEE) FFF) GGG) HHH)

III) JJJ) KKK) LLL) MMM) NNN) OOO) PPP) QQQ) RRR) SSS) TTT) UUU) VVV) WWW)

XXX) YYY) ZZZ) AAAA) BBBB) CCCC) DDDD) EEEE) FFFF) GGGG) HHHH) IIII) JJJJ) KKKK) LLLL)

MMMM) NNNN) OOOO) PPPP) QQQQ) RRRR) SSSS) TTTT) UUUU) VVVV) WWWW) XXXX) YYYY) ZZZZ) AAAAA)

BBBBB) CCCCC) DDDDD) EEEEE) FFFFF) GGGGG) HHHHH) IIIII) JJJJJ) KKKKK) LLLLL) MMMMM) NNNNN) OOOOO) PPPPP)

QQQQQ) RRRRR) SSSSS) TTTTT) UUUUU) VVVVV) WWWWW) XXXXX) YYYYY) ZZZZZ) AAAAAA) BBBBBB) CCCCCC) DDDDDD) EEEEEE)

FFFFFF) GGGGGG) HHHHHH) IIIIII) JJJJJJ) KKKKKK) LLLLLL) MMMMMM) NNNNNN) OOOOOO) PPPPPP) QQQQQQ) RRRRRR) SSSSSS) TTTTTT)

UUUUUU) VVVVVV) WWWWWW) XXXXXX) YYYYYY) ZZZZZZ) AAAAAAA) BBBBBBB) CCCCCCC) DDDDDDD) EEEEEEE) FFFFFFF) GGGGGGG) HHHHHHH) IIIIIII)

JJJJJJJ) KKKKKKK) LLLLLLL) MMMMMMM) NNNNNNN) OOOOOOO) PPPPPPP) QQQQQQQ) RRRRRRR) SSSSSSS) TTTTTTT) UUUUUUU) VVVVVVV) WWWWWWW) XXXXXXX)

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5. Find the center and radius of the circle whose equation is 1001422

yx .

6. Graph 2 22 25x y

Domain:

Range:

Another skill you will need for circles is writing the equation of a circle when given the

center and a point on the outside. This means you will need to find the radius length by

counting the spaces when you have a vertical or horizontal radius or by using the distance

formula or the Pythagorean Theorem if you have a diagonal radius. A piece of scrap graph paper

or at least a sketch always helps!

7) Write the equation of a circle centered at (5,-4) that is tangent to the x-axis.

8) Write the equation of a circle centered at (-3,6) that is tangent to the y-axis.

9) Write the equation of a circle centered at (4,7) that is tangent to the x-axis.

YYYYYYY) ZZZZZZZ) AAAAAAAA) BBBBBBBB) CCCCCCCC) DDDDDDDD) EEEEEEEE) FFFFFFFF) GGGGGGGG) HHHHHHHH) IIIIIIII) JJJJJJJJ) KKKKKKKK)

LLLLLLLL) MMMMMMMM) NNNNNNNN) OOOOOOOO) PPPPPPPP) QQQQQQQQ) RRRRRRRR) SSSSSSSS) TTTTTTTT) UUUUUUUU) VVVVVVVV) WWWWWWWW) XXXXXXXX)

YYYYYYYY) ZZZZZZZZ) AAAAAAAAA) BBBBBBBBB) CCCCCCCCC) DDDDDDDDD) EEEEEEEEE) FFFFFFFFF) GGGGGGGGG) HHHHHHHHH) IIIIIIIII) JJJJJJJJJ) KKKKKKKKK)

LLLLLLLLL) MMMMMMMMM) NNNNNNNNN) OOOOOOOOO) PPPPPPPPP) QQQQQQQQQ) RRRRRRRRR) SSSSSSSSS) TTTTTTTTT) UUUUUUUUU) VVVVVVVVV) WWWWWWWWW) XXXXXXXXX)

YYYYYYYYY) ZZZZZZZZZ) AAAAAAAAAA) BBBBBBBBBB) CCCCCCCCCC) DDDDDDDDDD) EEEEEEEEEE) FFFFFFFFFF) GGGGGGGGGG) HHHHHHHHHH) IIIIIIIIII) JJJJJJJJJJ) KKKKKKKKKK)

LLLLLLLLLL) MMMMMMMMMM) NNNNNNNNNN) OOOOOOOOOO) PPPPPPPPPP) QQQQQQQQQQ) RRRRRRRRRR) SSSSSSSSSS) TTTTTTTTTT) UUUUUUUUUU) VVVVVVVVVV) WWWWWWWWWW) XXXXXXXXXX)

YYYYYYYYYY) ZZZZZZZZZZ) AAAAAAAAAAA) BBBBBBBBBBB) CCCCCCCCCCC) DDDDDDDDDDD) EEEEEEEEEEE) FFFFFFFFFFF) GGGGGGGGGGG) HHHHHHHHHHH) IIIIIIIIIII) JJJJJJJJJJJ) KKKKKKKKKKK)

LLLLLLLLLLL) MMMMMMMMMMM) NNNNNNNNNNN) OOOOOOOOOOO) PPPPPPPPPPP) QQQQQQQQQQQ) RRRRRRRRRRR) SSSSSSSSSSS) TTTTTTTTTTT) UUUUUUUUUUU) VVVVVVVVVVV) WWWWWWWWWWW) XXXXXXXXXXX)

YYYYYYYYYYY) ZZZZZZZZZZZ) AAAAAAAAAAAA) BBBBBBBBBBBB) CCCCCCCCCCCC) DDDDDDDDDDDD) EEEEEEEEEEEE) FFFFFFFFFFFF) GGGGGGGGGGGG) HHHHHHHHHHHH) IIIIIIIIIIII) JJJJJJJJJJJJ) KKKKKKKKKKKK)

LLLLLLLLLLLL) MMMMMMMMMMMM) NNNNNNNNNNNN) OOOOOOOOOOOO) PPPPPPPPPPPP) QQQQQQQQQQQQ) RRRRRRRRRRRR) SSSSSSSSSSSS) TTTTTTTTTTTT) UUUUUUUUUUUU) VVVVVVVVVVVV) WWWWWWWWWWWW) XXXXXXXXXXXX)

YYYYYYYYYYYY) ZZZZZZZZZZZZ) AAAAAAAAAAAAA) BBBBBBBBBBBBB) CCCCCCCCCCCCC) DDDDDDDDDDDDD) EEEEEEEEEEEEE) FFFFFFFFFFFFF) GGGGGGGGGGGGG) HHHHHHHHHHHHH) IIIIIIIIIIIII) JJJJJJJJJJJJJ) KKKKKKKKKKKKK)

LLLLLLLLLLLLL) MMMMMMMMMMMMM) NNNNNNNNNNNNN) OOOOOOOOOOOOO) PPPPPPPPPPPPP) QQQQQQQQQQQQQ) RRRRRRRRRRRRR) SSSSSSSSSSSSS) TTTTTTTTTTTTT) UUUUUUUUUUUUU) VVVVVVVVVVVVV) WWWWWWWWWWWWW) XXXXXXXXXXXXX)

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10) Write the center-radius equation of a circle with a center at (-3, -6) and passes through

the point (-4, 8).

11) Write the center-radius equation of a circle with a center at (0,0) and passes through

the point (5,6).

12) Write the center-radius equation of a circle with a center at (0, 5) and passes through

the point (-3, 9).

13) Write the equation of the circle on the graph.

14) Write the equation of the circle on the graph.

Assume the scale is by ones.

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Lesson #29 Circles: General Form and Completing the

Square A2.A.47 Determine the center-radius form for the equation of a

circle in standard form

A2.A.24 Know and apply the technique of completing the square

Goal: Get a circle equation in standard form, such as 2 2 10 2 10 0x y x y , into center radius

form, 222rkyhx , so that you can find the center and radius.

Looking for patterns. Simplify the following squared binomials.

a. 2

3x

b. 2

5x

The 2 problems above are in the form: 2)( nx .

Describe the relationship between n and the second term in the trinomial.

Describe the relationship between the third term in the trinomial and n.

Factor. 2 6 9x x

2 10 25x x

If a circle equation is in standard form, we want to work backwards to center radius form so

that the x-terms and the y-terms squared binomials. In order to do so, we will use a technique

called completing the square so that the x-terms and y-terms can become squared binomials.

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1. Our first step is to organize the terms. At the same time move the constant to the

other side of the equation. Write “+__” after the x-terms and y-terms. 2 2

2 2

10 2 10 0

10 2 10

x y x y

x x y y

2. Find 2

band

2

2

b.

X terms: b=10, so 52

b and

2

252

b

Y terms: b=-2, so 12

b and

2

12

b

3. Add 2

2

b to the respective x and y terms. (This is the step where you are COMPLETING

THE SQUARE.) Add these numbers to the other side of the equation as well to keep it

balanced. 2 210 25 2 1 10 25 1x x y y

4. Factor the left side of the equation to put it in center radius form. This will always work

if you completed the square correctly.

2 2

5 5 ( 1)( 1) 36

5 1 36

x x y y

x y

In the example above, how can you still tell the original equation is a circle before you put it in

center radius form?

15) Convert this equation into center-radius form. State the coordinates of the center of

the circle and its radius.

To summarize: 2 2

2 2

2 2

2 2

10 2 10 0

10 2 10

10 25 2 1 10 25 1

( 5) ( 1) 36

x y x y

x x y y

x x y y

x y

Center:

Radius:

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16) Convert this equation into center-radius form. State the coordinates of the center of

the circle and its radius.

17) Convert this equation into center-radius form. State the coordinates of the center of

the circle and its radius. 2 2 16 20 90 0x y x y

You can also start with a center-radius equation and convert it to standard form by simplifying

and setting it equal to zero.

18) Convert 43422

yx to standard form.

19) State the equation of a circle in standard form which has a center at (5, -3) and a radius

of 9.

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Lesson #30: Finding the Roots of Higher Order Polynomials A2.A.50 Approximate the solution to polynomial equations of higher

degree by inspecting the graph

Determining the zeros

is the same as solving

the equation: 3 2 12 0x x x .

Based upon the last

lesson, this should

make sense. You are

finding when 3 21 12Y x x x

and 2 0Y intersect.

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You haven’t learned this, but we will do it

together for a preview.

Note: The roots, zeros, or

solutions are where the graph

crosses the x-axis. This is

usually what the question will ask

for.

Only the factors have the opposite signs.

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Lesson #31 – Solving Linear-Quadratic Systems Algebraically A2.A.3 Solve systems of equations involving one linear equation and one quadratic

equation algebraically. Note: This includes rational equations that result in

linear equations with extraneous roots

The past two lessons introduced the idea that graphs of functions can be used to solve

equations that you would normally solve algebraically.

Last year one of the things you learned how to do was solve linear-quadratic systems with a

graph.

Today we will be learning how to solve these systems of equations algebraically.

The big idea: In most cases, what you can solve with a graph you can also solve

with algebra and visa-versa.

Review: Below are graphs of linear-quadratic systems. You will see that there are a couple of

different possibilities for the number of solutions.

The equations will intersect in two

locations. Two real solutions.

The equations will intersect in one

location. One real solution.

The equations will not intersect.

No real solutions.

A question for next unit: What does it mean for a graph to have a real solution? What other

types of solutions would there be?

Solving Systems Algebraically A system of equations is two equations that work together to find a common solution. For that

reason I like to think of them as a team of equations. As you can see in the graph above, their

solutions are ordered pairs.

To solve systems algebraically we can use a method called substitution.

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Solving Systems by Substitution

2

8

2 8

y x

y x x 8y x

2

2

8 2 8

3 0

( 3) 0

0 3

x x x

x x

x x

x x

8

8 0

8

y x

y

y

8

8 3

5

y x

y

y

{(0,-8), (3,-5)}

1. Solve for x or y in one of the

equations. (The linear one is usually

easier.)

2. Substitute that value for x or y into the

other equation

3. Solve for the remaining variable.

4. Substitute the answer(s) into one of

the equations.

5. Solve for the other variable.

6. Write your solution(s) as ordered pairs

in roster notation.

1. Find the solution set of 2 4 3

1

x x y

y x algebraically.

Note: The quadratic relation below is a circle. We solve these types of equations in the same

way.

2. Solve algebraically: 2 2

5

97

y x

x y

Note: You can

check your

solution(s) to

systems of

equations by

storing both x

and y. Be sure to

check your

solution(s) in

both equations.

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In the following problem, there is a rational

expression. You would still solve this system the

same way, but you would have to make sure to

reject any solutions that are restricted values.

3. Solve for x: 21

9

3

y

x

y x

4. Solve for x algebraically. 2 2

1

25

y x

x y

5. Solve algebraically: 2 2 4 0

0

x y x

y x

6. Solve the following systems of

equations algebraically:

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Lesson #32 – Quadratic Inequalities in 2 variables A2.A.4 Solve quadratic inequalities in one and two variables,

algebraically and graphically

Graphing Quadratics Review Graph 2 5 2y x x

Steps:

1. Find the axis of symmetry,

2

bx

a.

2. Make a table of 7 points with the

axis of symmetry as the middle

value. (Even if the axis of

symmetry is a fraction, use

integers for the other x-values.)

3. Find the y-values using the

calculator table or by plugging

the x-values into the function.

4. Graph the points. Be sure to

label your graph. (All points do

not need to be on the graph).

Inequality Shading Review Sign Line Shading Example Sketches

>

Greater Than

<

Less Than

Greater Than or

Equal to

Less Than or Equal to

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