Unit 7 Notes - BABY SWINGS DANGEROUSsandtveit.weebly.com/uploads/6/9/9/3/69932021/unit_7...2 Unit 7 Calendar Part 1 – Graphing Quadratics Check your homework before class – Monday

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ALGEBRA 1 Unit 7 Name__________________________________ Graphing and Solving Quadratics Beginning of the Unit My goal for this unit is to earn __________ on the Unit 7 Test. Steps I will take to achieve this goal include:

1) ______________________________________________________________

2) _______________________________________________________________ 3) _______________________________________________________________ During the Unit - Strategies for Success

• Pay attention and do your best during class • Ask questions, work with your partner/group • Write down the Success Criteria • Use the Success Criteria when you get stuck • Do your homework – check your answers online, before coming to class – ask

questions on the ones you don’t understand • Complete the quiz and test reviews – ask questions on the ones you don’t understand • Come in for extra help if needed, right away! Don’t wait until the last minute.

End of the Unit 1) How long did you study for the test? What did you do to study for the test?

2) If you reached your goal that you set for yourself, what helped you do so? If you did

not reach your goal, why not?

3) What concepts do you still need extra practice on in order to understand them

better?

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Unit 7 Calendar Part 1 – Graphing Quadratics

Check your homework before class – www.sandtveit.weebly.com Monday Tuesday Wednesday Thursday Friday

4/9

Desmos Exploration

4/10 Testing ½ Day

Hours 1st-3rd Meet

Finish Section 1

Homework: 10.1 Worksheet

4/11 Testing ½ Day

Hours 4th-6th Meet

Finish Section 1

Homework: 10.1 Worksheet

4/12

Section 2 Graphing

2y ax bx c= + +

Homework Section 2 WS

4/12

Section 2 Graphing

2y ax bx c= + +


4/16 Intro to Intercept

Form Activity

4/17

Section 3 Intercept Form


4/18

Section 4 Solve by Graphing


4/19

Unit 7 Part 1 Test Review

4/20

Unit 7 Part 1 Test

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Unit 7 Success Criteria:

• I can graph a parabola given in standard form. Page(s) _____

• I can compare the transformation(s) of the graph of y = ax2 + c to y = x2. Page(s) _____

• I can graph a parabola given in intercept form. Page(s) _____

• I can solve a quadratic equation by graphing. Page(s) _____

• I can solve a quadratic equation by taking square roots. Page(s) _____

• I can solve a quadratic equation by completing the square. Page(s) _____

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• I can use completing the square to rewrite a quadratic equation in

vertex form. Page(s) _____

• I can graph a parabola given in vertex form. Page(s) _____

• I can solve a quadratic equation by using the quadratic formula. Page(s) _____

• I can use the discriminant to determine the number of solutions to a quadratic equation. Page(s) _____

• Given a pattern, I can determine whether a function is linear, exponential, or quadratic and I can write the corresponding equation. Page(s) _____

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Name: _______________________________ Unit 7: Quadratic Equations and Functions Section 1: Graph y = ax2 + c Quadratic Function – a nonlinear function that can be written in standard form, y = ax2 + bx + c, where a ≠ 0. Parabola – u-shaped graph of a quadratic function. Parent quadratic function – The most basic quadratic function, y = x2. Vertex – the lowest or highest point on a parabola. Axis of Symmetry – the line that passes through the vertex and divides the parabola into two symmetric parts.

Parent Function y = x2

x y

-2

-1

0

1

2

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Example 1: Make a table, graph and compare the graphs of y = 3x2 and y = x2

Compared to y = x2… Vertical: Expansion or Compression Reflection: Yes or No Translation: Up or Down or Neither Domain: __________________ Range: __________________

Example 2: Make a table, graph and compare the graphs of 2

41 xy = and y = x2

Vertical: Expansion or Compression Reflection: Yes or No Translation: Up or Down or Neither Domain: __________________ Range: __________________

y = 3x2

x y

-2

-1

0

1

2

2

41 xy =

x y

-2

-1

0

1

2

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Example 3: Make a table, graph and compare the graphs of 2

21 xy −= and y = x2


Comparing y=ax2 + c to y = x2 a is positive

_______________________________

a is negative

_______________________________

_______________________________

a < -1 or a > 1

_______________________________ -1 < a < 1

_______________________________

c is positive

_________________________________

________________________________

c is negative

________________________________

________________________________

2

21 xy −=

x y

-2

-1

0

1

2

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Example 4: Make a table, graph and compare the graphs of y = –3x2 + 5 and y = x2

Vertical: Expansion or Compression Reflection: Yes or No Translation: Up or Down or Neither Domain: __________________ Range: __________________ Example 5: Make a table, graph and compare the graphs of y = –3x2 + 5 and y = x2


y = –3x2 + 5

x y

-2

-1

0

1

2

y = ½x 2 – 1

x y

-2

-1

0

1

2

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Section 2: Graph y = ax2 + bx + c PROPERTIES OF THE GRAPH OF A QUADRATIC FUNCTION To graph a parabola in the form y = ax2 + bx + c : Step 1: Determine if the parabola opens up or down.

Ø If the a-value is _______________ , it opens _______________.

Ø If the a-value is _______________ , it opens _______________.

Step 2: Find the axis of symmetry (the mirror of the parabola).

Ø Step 3: Find and plot the vertex (middle point of the parabola).

Ø Step 4: Find more points on the graph by getting the calculator table. Plot the points.

(Center the vertex)

Vertex

x

y

Other Important Information About Parabolas… Minimum valueMinimum value Maximum valueMaximum value yy -- InterceptIntercept RootsRoots

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Example 1: Graph y = y = -- xx22 + 4x + 4x –– 11 Step 1Step 1: Determine whether the parabola opens up or down Step 2Step 2: Find and draw the axis of symmetry: Step 3Step 3: Find and plot the vertex. Step 4Step 4: Find more points on the graph. Choose x-values less than and

greater than the vertex.

Vertex

x

y

Domain: ___________ Range: __________ Example 2: Graph y = 2xy = 2x22 –– 8x + 78x + 7 Step 1Step 1: Determine whether the parabola opens up or down Step 2Step 2: Find and draw the axis of symmetry: Step 3Step 3: Find and plot the vertex. Step 4Step 4: Find more points on the graph. Choose x-values less than and

greater than the vertex. Domain: ___________ Range: __________

Vertex

x

y

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Example 3: Graph y = 3xy = 3x22 –– 6x + 26x + 2 Step 1Step 1: Determine whether the parabola opens up or down Step 2Step 2: Find and draw the axis of symmetry: Step 3Step 3: Find and plot the vertex. Step 4Step 4: Find more points on the graph. Choose x-values less than and

greater than the vertex. Domain: ___________ Range: __________

Vertex

x

y

Example 4: The cables between two telephone poles can be modeled by the equation y = 0.0024x2 – 0.1x + 24, where x and y are measured in feet. To the nearest foot, what is the height of the cable above the ground at its lowest point?

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Section 3 - Graph Quadratic Functions in Intercept Form Exploration: Each group will receive a quadratic equation to examine. You will create a poster with the following information. If there are only three group members, one person should do the tasks for group members #1 and #4. Group Member #1

• Determine a = _______ b = _______ c = ________ • Does the parabola open up or down?

• Determine the axis of symmetry: abx2−

=

Group Member #2 • Enter the equation into the calculator • Create a table

vertex

Group Member #3

• Graph the equation • Identify the x-intercepts

Group Member #4

• Solve the equation by factoring When you’re done, put a star or circle your

• Factored form • Solutions • X-intercepts

Poster Summary

Equation Factored Form Solutions x -intercepts y = x2 – 6x + 8

y = -x2 + 8x – 12

y = x2 – 2x – 8

y = -x2 + 4x

y = x2 + 8x + 7

y = -x2 – 2x + 3

y = x2 – 4

y = x2 – 6x + 5

Why is the factored form referred to as intercept form? What is the relationship between the solutions of the equation and the x-intercepts?

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

Ø Step 1: Find the x-intercepts by setting the factors equal to zero.

Ø Step 2: The Axis of Symmetry is halfway between the roots.

Ø Step 3: Find and plot the vertex.

Ø Opens Up if a-value is _________________.

Opens Down if a-value is _________________. Example 1: Graph y = y = -- (x + 1)(x (x + 1)(x –– 5)5) Step 1Step 1: Identify and plot the x-intercepts Step 2Step 2: Find and draw the axis of symmetry: Step 3Step 3: Find and plot the vertex. Domain: ___________ Range: __________

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Example 2: Graph y = (x + 1)(x y = (x + 1)(x –– 3)3) Step 1Step 1: Identify and plot the x-intercepts Step 2Step 2: Find and draw the axis of symmetry: Step 3Step 3: Find and plot the vertex. Domain: ___________ Range: __________ Example 3: y = 2x2 – 8 Step Step 1: Rewrite the quadratic function in intercept form.

Step 2Step 2: Identify and plot the x-intercepts Step 3Step 3: Find and draw the axis of symmetry: Step 4Step 4: Find and plot the vertex. Domain: ___________ Range: __________

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Section 4: Solve Quadratic Equations by Graphing

NUMBER OF SOLUTIONS OF A QUADRATIC EQUATION

Two Solutions One Solution No Solutions (Unfactorable)

Example 1: Solve xx22 + 4x = 5+ 4x = 5 by graphing. Step 1: Step 1: Put equation into standard form.

Step Step 22: Plug equation into calculator. Locate the intercepts (y = 0) and center the vertex.

Step Step 33: Factor and solve the equation to check your roots.

Vertex

x

y

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Example 2: Solve xx22 –– 4x = 4x = --44 by graphing. Step 1: Step 1: Put equation into standard form.


Step Step 33: Factor and solve the equation to check your roots. Example 3: Solve xx22 + 8 = 2x+ 8 = 2x by graphing. Step 1: Step 1: Put equation into standard form.


Step Step 33: Factor and solve the equation to check your roots.

Vertex

x

y

Vertex

x

y

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Equation Solving Inquiry

What similarities do you notice among the equations?

What differences do you notice among the equations?

Solve each equation to the best of your ability. 1. 6(2x + 10) = 5x + 25

2. 6x2 + 5x – 4 = 0

3. 2x2 – 32 = 0

4. (x – 1)2 = 9

5. ( )4 7 123

x− =

6. x2 + 36x + 11 = 0

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Section 5: Use Square Roots to solve Quadratic Equations Step 1: Get the squared term all by itself on one side of the equal sign. x2 = (number) Step 2: To “undo” the square term, we have to do the inverse, which is to ________________.

If the (number) is positive, you will have ________ solution(s). )(numberx ±=

If the (number) is zero, you will have ________ solution(s). 0±=x If the (number) is negative, you will have ________ solution(s). Ø Step 3: Solve the remaining equations for the variable (if necessary). Example 1: Solve the Equation. a. z2 – 5 = 4 b. 25k2 = 9 c. 4x2 + 3 = 23 d. 6g2 + 1 = 19 e. 5(x + 1)2 = 30 f. 3(m – 4)2 = 12

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Example 2: During an ice hockey game, a remote-controlled blimp flies above the crowd and drops a numbered table-tennis ball. The number of the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air.

Example 3: You drop your glasses from a balcony 18 feet above your den on to a table that is 3 feet above the ground. How long are your glasses in the air?

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Section 6: Solve Quadratic Equations by Completing the Square Use a box to square the binomial. 1. (x + 3)2 2. (x + 4)2 3. (x – 7)2

What patterns do you notice? Complete the Square: How can we reverse the procedure? Find the number that ‘completes the square.’ 1. x2 + 4x + _____ = (x + ___ ) 2 2. x2 – 16x + ____ = (x - ___)2

3. x2 + 12x _____ = (x + ___ )2 4. x2 – 22x + ____ = ( x - ___ )2

Let’s Practice! Find the value of ‘c’ such that each expression is a perfect square trinomial. Then write the expression as the square of a binomial. 5. x2 + 18x + c 6. r2 – 4r + c 7. p2 – 30p + c

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Completing the Square Step 1: Put the expression into this form:

x2 + bx + _____ = (number) or x2 – bx + _____ = (number) Step 2: To fill in the blank, take half of “b”, square it and add it to both sides.

x2 + bx + 2

2⎟⎠

⎞⎜⎝

⎛ b = (number) +2

2⎟⎠

⎞⎜⎝

⎛ b or x2 – bx + 2

2⎟⎠

⎞⎜⎝

⎛ b = (number) + 2

2⎟⎠

⎞⎜⎝

⎛ b

Step 3: Factor the left side into a perfect square, simplify the right side. Step 4: Square root both sides to solve for x. Example 1: Solve the quadratic equation a. x2 + 6x = -5 b. r2 – 8r = 9 c. x2 + 6x = 7 d. d2 – 24 = 5d

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Example 2: Solve the equation by completing the square. a. 4m2 – 16m + 8 = 0 b. 5s2 + 60s + 125 = 0 c. 3x2 + 12x – 18 = 0 d. 2k2 + 20k = 8 Example 5: You are designing a herb garden with a uniform border of ornamental grass around it as shown below. Your design includes 12 square feet for the herb garden. Find the width of the grass border to the nearest inch.

x

x

x

x 16 ft2

6 ft

6 ft

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Section 7: Graph Quadratic Functions in Vertex Form Sometimes quadratic functions are written:

y = a(x – h)2 + k This form of a quadratic is known as VERTEX FORM. Let’s see if we can figure out why. 1. y = (x – 3)2 + 4 Rewrite the given equation in standard form: y = ax2 + bx + c Graph the equation by finding the axis of symmetry and creating a table. vertex

2. y = (x + 1)2 – 2 Rewrite the given equation in standard form: y = ax2 + bx + c Graph the equation by finding the axis of symmetry and creating a table. vertex

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3. y = -(x + 2)2 – 3 Rewrite the given equation in standard form: y = ax2 + bx + c Graph the equation by finding the axis of symmetry and creating a table. vertex

4. y = 2(x + 3)2 + 1 Rewrite the given equation in standard form: y = ax2 + bx + c Graph the equation by finding the axis of symmetry and creating a table. vertex

Why do you think y = a(x – h)2 + k is called vertex form? __________________________________________________________________________________________

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Vertex Form – ( ) khxay +−= 2

Ø Vertex:

Ø Axis of Symmetry:

Ø Up if: Down if: Example 1: Graph a quadratic function in vertex form y = -(x + 2)2 + 3 Step 1: Step 1: Identify the values of a, h, and k.

Step 2Step 2: Identify and draw the axis of symmetry. Step 3Step 3: Find and plot the vertex (h, k). Step 4Step 4: Find more points on the graph. Choose x-values less than and


Vertex

x

y

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Example 2: Write the function in vertex form (complete the square) and graph the function. y = x 2 – 8x + 11 Step 1: Step 1: Write the function in vertex form by completing the square. Step 2: Step 2: Identify the values of a, h, and k.



Vertex

x

y

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Example 3: Write the function in vertex form (complete the square) and graph the function. a. y = x 2 – 6x + 5 Step 1: Step 1: Write the function in vertex form by completing the square. Step 2: Step 2: Identify the values of a, h, and k.



Vertex

x

y

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Section 8: Solve Quadratic Equations by the Quadratic Formula The Quadratic Formula – Used to solve quadratics equations in standard form.

Standard Form: Quadratic Formula:

02 =++ cbxax aacbbx

242 −±−

=

Example 1: Use the quadratic formula to solve a. 2x2 – 5 = 3x b. 2x2 + 7x = 9 c. 3x2 – 1 = x

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Example 2: A crabbing net is thrown from a bridge, which is 35 feet above the water. If the net’s initial velocity is 10 feet per second, how long will it take the net to hit the water? Example 3: For the period 1990-2003, the number of book titles published by a small publishing company can be modeled by the function y = 0.5x2 + 4x + 19, where x is the number of years since 1990. In what year did the company publish 80 books?

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Section 9: Interpret the Discriminant Discriminant – Can determine how many solutions a quadratic equation has.

aacbbx

242 −±−

=

USING THE DISCRIMINANT OF ax2 + bx + c = 0

Value of the discriminant Number of solutions Graph of

y = ax2 + bx + c = 0

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Example 1: Find the discriminant and determine the number of solutions a. x2 – 3x – 2 = 0 b. 3x2 + 7x = -5 c. -x2 – 9 = 6x d. 3x2 + 2 = 0 e. 2x2 + 8x + 8 = 0 f. x2 + 2x = 1

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Section 10: Compare Linear, Exponential, and Quadratic Models Linear Function Exponential Function Quadratic Function y = y = y =

Example 1: Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function. a. (-2, 7), (-1, 1), (0, -1), (1, 1), (2, 7) b. (-2, 4), (-1,2), (0, 1), (1, ½), (2, ¼)

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Differences and ratios in a table Linear equation – ________________________________________________________________________ Exponential function – ____________________________________________________________________ Quadratic Function – ____________________________________________________________________ Example 2: Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write the equation for the function a.

x -2 -1 0 1 2

y 4 1 0 1 4

b.

x -2 -1 0 1 2

y -12 -8 -4 0 4

c.

x -2 -1 0 1 2

y 0.25 .05 1 2 4

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Example 3: The table shows the cost to run an as in a magazine. Tell whether the data can be modeled by a linear function, an exponential function, or a quadratic function. Then write the equation for the function. Number of

lines, x Total cost,

y

4 $10.40

5 $12.25

6 $14.10

7 $15.95

8 $17.80

9 $19.65

Example 4: The table shows the distance a clock’s pendulum swings after certain amounts of time. Tell whether the data can be modeled by a linear function, an exponential function, or a quadratic function. Then write the equation for the function.

Time in seconds,

x

Distance in feet,

y

1 0.82

2 3.28

3 7.38

4 13.12

5 20.5

Documents

Unit 7 Notes - BABY SWINGS DANGEROUSsandtveit.weebly.com/uploads/6/9/9/3/69932021/unit_7...2 Unit 7 Calendar Part 1 – Graphing Quadratics Check your homework before class – Monday