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Honors Algebra 2 – Unit 4 Notes

Day 1 – Graph Quadratic Functions in Standard Form

GOAL: Graph parabolas in standard form y = ax2 + bx + c

Quadratic Function - Parabola -

Vertex - Axis of symmetry -

Minimum and maximum value -

PARENT FUNCTION FOR QUADRATIC EQUATIONS

Equation: Axis of symmetry:

Vertex: Min/Max Value:

Domain: Range:

Increasing: Decreasing:

PROPERTIES OF THE GRAPH y = ax2 + bx + c

The graph opens up if a 0 and down if a 0.

The graph is vertically stretched if |a| 1 and vertically shrunk if |a| 1.

The axis of symmetry is x = and the vertex has an x coordinate of .

The y-intercept of the graph is . So the point (0, c) is on the parabola.

Graph a function in the form of y = ax2 + bx + c.

1. y = –x2 + 4x – 3

Identify a, b, c:

Find the vertex:

Draw the axis of symmetry:

Identify the y-intercept:

Complete the table using other x-values near the vertex:

Draw the parabola.

MINIMUM AND MAXIMUM VALUES

If a > 0, then there is a value.

If a < 0, then there is a value.

Find the minimum or maximum value.

2. y = –3x2 + 12x – 6 3. f(x) = 7x2 + 21x + 8

4. Find the vertex of the function given. y = 1

3x2 + 3x

Graph the following functions, then find all the requested info.

5. y = x2 – 6x + 5

Compare this graph with the parent function, and then state the domain and range.

Comparison:

Domain: Range:

Increasing: Decreasing:

Positive: Negative:

Minimum or Maximum (what is it?):

6. g(x) = 2 x2 + 2

Compare this graph with the parent function, and then state the domain and range.

Comparison:

Domain: Range:

Increasing: Decreasing:

Positive: Negative:

Minimum or Maximum (what is it?):

Day 2 – Graph Absolute Value Equations (Intro to h, k translations)

GOAL: Graph absolute value functions in (h,k) form

Absolute Value Review

1. 4 8 = 2. 2 6 12 = 3. 0 3 =

Absolute Value Functions

Vertical Stretch /Shrink:

Reflection:

Translation:

a always represents the slope of the _______________________.

h

k

y a x h k | |

Graph the following using transformations.

1. 4 2y x 2. 2 5f x x | | 3. 5 2f x x | |

4. 2 3y x 5. 0 5 2 3f x x . | |

6. Write the equation of the absolute value function that is translated left 7 units and down 4.

7. Given the equation y x 1 4 , if this graph was vertically stretched by a factor of 5, translated down 4

units and right 3, what is the equation of the new graph?

Day 3 – Graph Quadratic Functions in Vertex Form

GOAL: Graph parabolas in vertex form

Vertex Form –

VERTEX FORM OF A QUADRATIC FUNCTION: y = a(x – h)2 + k

h translates the graph ____________ or _____________

k translates the graph ____________ or _____________

a vertically _____________ or _____________ the graph and if a is negative

State the vertex of each.

1) y = (x – 8)2 + 6 2. y = -9(3x + 1)2 – 7 3) y = 3x2 + 5

versus…

4) y = 2x2 + 2x 5) y = -7x2 – 21x

Graph the following quadratic functions in vertex form. Label the vertex and axis of symmetry.

6. 2( 1) 2y x 7. 22( 3) 4y x 8. 21( ) 3

2f x x

Day 4 – Graph Quadratic Functions in Vertex Form and Standard Form

GOAL: Graph parabolas in standard and vertex form and determine the end behavior of functions.

Write the following functions in vertex form.

1. y = x2 +14x + 44 2. f(x) = x2 – 12x + 22 3. y = 2x2 +24x + 25

Graph the following quadratic functions. Label the vertex, the axis of symmetry, and state the domain

and range.

4. 2 4 5y x x 5. 23 6 1y x x

Vertex: Vertex:

Axis of Sym: Axis of Sym:

Domain: Domain:

Range: Range:

End behavior for each graph above –

Using a graphing calculator, state the end behavior of each.

6. f(x) = 1

𝑥 7. f(x) = - x3 + 1 8. y =

2

𝑥2

Graph the following quadratic functions. Label the vertex, the axis of symmetry, state the domain

and range, and identify the end behavior of the graphs.

9. 2

3 2f x x Vertex:

Axis of Symmetry:

Domain: Range:

Increasing: Decreasing:

Positive: Negative:

Zeroes:

Minimum or Maximum (what is it?):

( )f x _ __ as x and ( )f x as x

10. 212 3

2g x x x Vertex:

Axis of Symmetry:

Domain: Range:

Increasing: Decreasing:

Positive: Negative:

x-intercepts:

Minimum or Maximum (what is it?):

( )g x as x and ( )g x as x

Day 5 – Applying the h, k Format to ANY Function

𝑦 = ±𝑎(𝑏(𝑥 − ℎ)) + 𝑘

Without graphing, identify the name of the parent function and the general shape of the parent

function graph. Then describe how the graph of the given function will be transformed from the

parent graph by stating the vertex.

1. 𝑦 = 7(𝑥 + 2)2 − 3 2. 𝑦 = −0.6|𝑥 + 5| + 9

3. 𝑦 =4

3(2𝑥 − 3) + 4

4. 𝑦 = −3𝑥2

5. 𝑦 =

2

3(3𝑥 − 2)3 + 7

6. 𝑦 = 4|𝑥 + 5| + 9

+

–

0 < |𝑎| < 1 0 < |𝑏| < 1

|𝑎| = 1 |𝑏| = 1

|𝑎| > 1 |𝑏| > 1

𝑥 − ℎ

𝑥 + ℎ

+𝑘

−𝑘

Together h and k create the “vertex” point

(h, k)

The symbols around x

determine the parent function

Describe the transformation upon the graph of each given function.

7. 2g x 8. 1h x 9. 6 4f x

10. 6 8f x 11. 3h x 12. 1

( 1) 72

g x

Use the notation above to communicate the transformations of the graphs below (red is the original).

13. 14.

Day 6 – Graph Quadratic Functions Intercept Form

GOAL: Graph parabolas in intercept form and transform equations into intercept form.

Intercept form –

CHARACTERISTICS OF A GRAPH IN INTERCEPT FORM: y = a(x – p)(x – q)

The x-intercepts are and .

The axis of symmetry is halfway between (p, 0) and (q, 0). It has an equation 2

x .

The same properties of a apply.

Graph the following quadratic functions in intercept form. Label the vertex, the axis of symmetry, state

the domain and range, and identify the end behavior of the graphs. Also state on what x intervals the

function is increasing/decreasing.

1. 4 2f x x x 2. 1 5g x x x

Vertex: Vertex:

Axis of Symm: Axis of Symm:

Domain: Domain:

Range: Range:

( )f x as x and ( )f x as x ( )g x as x and ( )g x as x

Increasing: Increasing:

Decreasing: Decreasing:

3. 2 2 3f x x x 4. 22 8 6g x x x

Vertex: Vertex:

Axis of Symm: Axis of Symm:

Domain: Domain:

Range: Range:

( )f x as x and ( )f x as x ( )g x as x and ( )g x as x

Increasing: Increasing:

Decreasing: Decreasing:

Day 7 – Changing Forms of a Quadratic Function

GOAL: Rewrite quadratic function in various forms.

Write the following functions in standard form.

1. f(x) = –5(x + 2)2 + 8 2. y = 4(x – 3)2 – 10 3. Has a vertex at (5,0) and is

vertically stretched by a factor of 2.

Write the following functions in standard form.

4. y = 3(x + 2)(x – 5) 5. f(x) = –3(x – 7)(x + 6) 6. Has zeros of –8 and 3 and is

reflected across the x-axis.

Write the following functions in intercept form.

7. f(x) = (x – 4)2 – 1 8. Has a vertex at (–2, –4) and whose end behavior

approaches positive infinity.

Write the following functions in vertex form.

9. y = (x – 7)(x – 1) 10. Has zeros of –1 and –5, is vertically shrunk by a

factor of 1/2, and is reflected across the x-axis .

11. Vertex is (5, -6) and goes through the point (2, 12)

Day 8 – Application of Quadratics

Goal: Apply Quadratic Models

1. Mr. Coker and Mrs. Revilleza have recently taken up the game of tennis. Mr. Coker lobs his meanest shot

to Mrs. Revilleza, and it can be modeled by the function 21

f (x) x 3 82

where x is the horizontal

distance (in feet) from where Mr. Coker hit the ball and f(x) is the height of the ball (in feet) above the court.

A) Graph the function and state the B) In the context of this problem…

domain and range. What does the domain stand for?

What is it?

C) In the context of this problem…

What does the range stand for?

What is it?

D) What is the maximum height of the tennis ball?

Domain: Range:

E) From what height is the tennis ball hit? F) How far away from Mr. Coker does the tennis

ball reach its maximum height?

G) If Mrs. Revilleza is blinded by the sun, and does not return the fierce lob hit by Mr. Coker, how far does Mr.

Coker’s shot go?

H) If Mr. Coker is six feet away from the net, which has a height of 3 feet, will the ball even clear the net?

Explain your reasoning.

I) What does f(1) = 6 represent in terms of this problem?

J) What is the height of the ball after it has traveled 5 feet horizontally?

2. On the court next to Mr. Coker, Mr. Pacheco and Ms. Simi are also partaking in a feisty game of tennis. Ms.

Simi’s latest shot can be modeled by the function 2h(t) 1.5t 3t 4.5 where t is the time (in seconds) that

the ball is in the air and h(t) is the height of the ball (in feet) above the court.

A) Graph the function and state the B) In the context of this problem…

domain and range. What does the domain stand for?

What is it?

C) In the context of this problem…

What does the range stand for?

What is it?

D) At what time does the tennis ball reach its

maximum height?

Domain: Range:

E) What is the maximum height? F) From what height is the tennis ball hit?

G) If Mr. Pacheco slips and does not return the shot hit by Ms. Simi, how far does the ball go?

H) So if the ball is not hit, how long was it in the air for when it hits the ground?

I) What does h(2.5) = 2.625 represent in terms of this problem?

3. Both of the games above though pale in comparison to the match taking place between Mr. Brandt and Ms.

Parish on Centre Court. Ms. Parish zings a shot in Mr. Brandt direction, and he dives to barely hit it a split

second before it hits the ground. If Mr. Brandt’s shot is modeled by the following function

y .0025x(x 78.9) where x is the horizontal distance (in feet) from where Mr. Brandt’s hit the ball and y is

the height of the ball (in feet) above the court, does Mr. Brandt; shot land inside the court if he is 78 feet away

from Ms. Parish’s baseline?

4. The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and

are connected by suspension cables. Each cable can be modeled by the function 2714007000

1)(

2 xxf ,

where x is the distance from one tower in feet and f(x) is the height of the cable above the roadway. What is the

distance between the two towers?

5. According to the function in problem #4, how far above the roadway is the cable at its lowest point?

6. The path of a kicked soccer ball can be modeled by the function )80(02.0 xxy , where x is the

horizontal distance in feet and y is the height of the ball in feet. How far does the soccer ball travel in the air?

7. Using the information about the soccer ball above, what is the maximum height of the ball?