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Unit 2-2: Writing and Graphing Quadratics
NOTE PACKET Name:__________________Period______ Learning Targets:
Unit 2-1 12. I can use the discriminant to determine the number and type of solutions/zeros.
Modeling with
Quadratic Functions
1. I can identify a function as quadratic given a table, equation, or graph.
2. I can determine the appropriate domain and range of a quadratic equation or event.
3. I can identify the minimum or maximum and zeros of a function with a calculator.
4. I can apply quadratic functions to model real-life situations, including quadratic regression models from data.
Graphing
5. I can graph quadratic functions in standard form (using properties of quadratics).
6. I can graph quadratic functions in vertex form (using basic transformations).
7. I can identify key characteristics of quadratic functions including axis of symmetry, vertex, min/max, y-intercept, x-intercepts, domain and range.
Writing Equations of Quadratic Functions
8. I can rewrite quadratic equations from standard to vertex and vice versa.
9. I can write quadratic equations given a graph or given a vertex and a point (without a calculator).
10. I can write quadratic expressions/functions/equations given the roots/zeros/x-intercepts/solutions.
11. I can write quadratic equations in vertex form by completing the square.
Revisit Applications
4R. I can apply quadratics functions to real life situations without using the graphing calculator.
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Unit 2-2 Writing and Graphing Quadratics Worksheets Completed Date LTs Pages Problems Done Quiz/Unit Test Dates(s) Date LTs Score Corrected Retake
Quiz Retakes Dates and Rooms
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Modeling Quadratic Functions Date: ____________ After this lesson and practice, I will be able to …
¨ identify a function as quadratic given a table, equation, or graph. (LT 1) ¨ determine the appropriate domain and range of a quadratic equation or event. (LT 2) ¨ identify the minimum or maximum, and zeros of a function with a calculator. (LT 3) ¨ apply quadratic functions to model real-life situations, including quadratic regression models. (LT 4)
--------------------------------------------------------------------------------------------------------------------------------------------------- Let’s review the definition of a quadratic function …
Quadratic Function: A function that can be written in the form ____________________ where ____________. The domain of a quadratic function is _________________________________. Standard Form of a Quadratic Function:
Example 1: Determine whether each function is quadratic or not. If so, write it in standard form. a. y = x(x-‐5) b. c. y= x2 – (5x + 2 + x2) d. f(x) = (2x+3)(x-‐7) Properties of Quadratic Functions Let’s graph the parent function for a quadratic : 𝑦 = 𝑥! The shape of the graph is called a ____________________. The point at which the y value of the graph comes to a maximum or minimum is called the ___________________. It is the ordered pair _______________________. The vertical line line that passes through the vertex of the parabola is called the __________________________. It has the equation ___________________________.
x y -‐3 9 -‐2 4 -‐1 1 0 0 1 1 2 4 3 9
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Example 2: Suppose a quadratic function as a vertex of (-‐3,-‐5) and contains the points (-‐4,-‐3) and (-‐2,3). Identify the equation for the axis of symmetry and use it to help you sketch a graph of the function. Then identify the domain and range. Example 3: Suppose a quadratic function as a vertex of (5,6) and contains the points (7,2) and (2,-‐3). Identify the equation for the axis of symmetry and use it to help you sketch a graph of the function. Then identify the domain and range. The places where the graph crosses the x -‐axis are called: Using Your Calculator (LT 3) In some situations, you want to quickly approximate the __________ and the _____________________/____________________of a quadratic equation. Thankfully, your graphing calculator is very helpful in accomplishing this feat. Example 4: Approximate the zeros and min or max of the equation 3x2 + 5x = 6+y using a graphing calculator. Round to four decimal places. Approximate the zeros Note: This method will not work if the zero is a min or max. 1) Put the equation you want to graph in standard form. 2) Enter the equation in your calculator as Y1 =. Press GRAPH. (Adjust your window if needed. ZOOM ZStandard gives 10 by 10 window) 3) Press 2nd TRACE, then press 2: ZERO. 4) Left Bound? Move your cursor just to the “left” of the first point of intersection. Press ENTER. 5) Right Bound? Move your cursor just to the “right” of the first point of intersection. Press ENTER. 6) The screen will show “Guess”. Press ENTER again. 7) The bottom of the screen will say X=____ Y=0. The x value is the zero. The first zero in this example is _________. 8) Repeat steps 3-7 to obtain the second root. The second zero is __________.
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If you are having trouble using the ZERO function try this to approximate zeros: 1) Put the equation you want to graph in standard form. 2) Put equation into Y1 and y=0 into Y2. 3) Press 2nd TRACE, then press 5: INTERSECT 4) First curve? Move your cursor close to the point of intersection. Press ENTER. 5) Second Curve? If needed move your cursor close to the point of intersection. Press ENTER. 6) The screen will show “Guess”. Press ENTER again. 7) The bottom of the screen will say X=____ Y=0. The x value is the zero. The first zero in this example is _________. 8) Repeat steps 3-7 to obtain the second root. The second zero is __________. Approximate the min or max 1) Press 2nd TRACE, then press MIN or MAX (depending on the shape of your parabola). 2) Move your cursor just to the “left” of the min or max. Press ENTER. 3) Move your cursor just to the “right” of the min or max. Press ENTER. 4) The screen will show “Guess”. Press ENTER again. 5) The bottom of the screen will say X=____ Y =_______ The y value is the min or max. The x value is where the min or max is occurring. The min or max in this example is _________ at _______. Example 5: Approximate the zeros and min or max of the equation below using a graphing calculator. Round to four decimal places.
a) −2x2 +10x−1= y b) f (x)= x2 +8x+16 Zero(s) :_________________________ Zero(s) :_________________________
Min Max is ________ at __________ Min Max is ________ at __________
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Applications of Quadratic Functions (LT 4) Quadratic functions have many applications to real-‐world scenarios. For example, quadratics can be used to model the height of a projectile over time. They can also be used to help businesses predict revenue. Example 6: The height of a punted football can be modeled by h=−0.01x2 +1.18x+2 . The horizontal distance in feet from the punter’s foot is x, and h is the height of the ball in feet. Calculate how far the punter kicked the ball using your graphing calculator. ____________________ feet ___________________ yards Example 7: Smoke jumpers are firefighters who parachute into areas near forest fires. A jumper’s height during free fall (before the parachute opens) is modeled by
!!h(t)= −16t2 +h0 where t is time in seconds and !!h0 is the initial height. If a jumper’s plane is
1500 feet off the ground when he jumps, how long is he in free fall before his parachute opens at 1200 feet? ______________ Example 8: An apartment complex has 100 two-‐bedroom units. The monthly profit in dollars realized from renting out the apartments is given by: P(x) = -‐10x2 + 1500x – 20,000 a. How many units must be rented out to maximize profits? ____________________ b. What is the maximum profit realizable? ____________________ c. How many apartments must be rented before a profit is made? ___________________ Example 9 : A bakery knows from market research that -‐6p + 180 models the number of pies sold on average day, where p is the price of a pie in dollars. a. Write an equation to represent the revenue the bakery can expect on an average day. R(p) = _______________________________ b. What price will maximize the revenue? ____________________ c. What is the maximum revenue? ____________________
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Example 10 If Jimmy Fallon dropped a T.V. from a height of 1500 feet, how long would it take to hit the ground? How long would it take to land on a building 50 feet tall. Use the formula: h(t) = -‐16t2 + s h(t) = ____________________ h = height when the object landed. If it landed on the ground, then h = ______.
If it landed on the top of a 50 ft. building, then h = ______. t = the time it takes for the object to fall. s = height from which the object is dropped or…. where the object started! _____________ time to ground ______________ time to building ______________ Example 11: The data in the table represents the amount of rainfall for each month, in inches. a. Graph a scatter plot of the data in your calculator. What type of function best represents the data? -‐ Enter your data (2ND STAT, Edit…) -‐ Turn on Stat Plot 1 (2nd, STAT PLOT) -‐ Graph the data (adjust window if needed) b. Find the ____________ regression line that models the data using your calculator.
-‐ STAT, calc, QuadReg -‐ Look at R2 value to see if it is a good fit
R(x) = __________________________________ Type equation into Y1 or use directions below: Stat Calc QUADReg VARS Y-‐VARS Function Y1 Now Graph and see how line matches scatterplot c. Using your equation, predict the rainfall in November. R(11) =
Month Rainfall (inches)
Jan 5.7 Feb 4.2 Mar 3.8 Apr 2.4 May 1.7 Jun 1.6 Jul 0.8 Aug 1 Sep 1.8 Oct 2.1 Nov ??? Dec 5.4
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Graphing with Properties of Quadratics Date: ____________ After this lesson and practice, I will be able to …
¨ graph quadratic functions in standard form (using properties of quadratics). (LT 5) ¨ identify key characteristics of quadratic functions including axis of symmetry, vertex, min/max, y-
intercept, x-intercepts, domain and range (LT 7) --------------------------------------------------------------------------------------------------------------------------------------------------- One of the most important things we will do this year is graph functions. The simplest (but not always easiest way) of graphing is to make a table … Example 1: Graph the quadratic function by making a table, then write the axis of symmetry, the domain, and the range: y = -‐2x2 + 3
Now let’s explore some easier ways of graphing. Start with the parent quadratic function: ________________
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Go to www.desmos.com and click Start Graphing. In the first blank, type in the parent function (use the carrot key ^ for the squared part of the equation). You should see a red parabola appear with its vertex at (0,0). Follow the directions below and fill in the blanks. In the box where you just wrote the parent function, add an “a” before the x. You should see a blue button appear next to the words “add slider”. Click the blue button. You should now be able to change the value of a by using the slider. Experiment with the slider, then fill in the blanks … When 𝑎 > 0, the parabola opens __________. When 𝑎 < 0, the parabola opens __________. Go back to the parent function and add “+bx”. The first box should now read: 𝑦 = 𝑎𝑥! + 𝑏𝑥. Add a slider for b like you did with a, then experiment with moving b around. In Box 4, add the following equation: 𝑥 = − !
!! . The fraction will automatically form when you type in “/”.
You should see a different colored line appear on the graph. Now experiment with the a and b sliders. 𝑥 = − !
!! is the ___________________________________. (It is an equation x = ______)
So the x-‐coordinate of the vertex is − !
!! and the y-‐coordinate is what you get when you plug in − !
!!.
In other words, the vertex is: − !!! , 𝑓(− !
!! ) (It is an ordered pair.)
Finally, add “+c” to the parent function in Box 1. It should now read: 𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐. Add a slider for c, then experiment with moving a, b, and c. The point (0, c) is the __________________________. These properties make graphing much easier! Graph the following functions on your calculator
y = 3x2 – 5 y = 0.5x2 + 2x + 3 y = -‐x2 + 3x + 7 y = -‐0.2x2 – x + 8
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Example 2: Graph y = x2 – 2x -‐ 3 (Identify a, b, and c? a= __________ b= ___________ c=___________ Steps to graphing quadratic functions: 1. Find the equation of the axis of symmetry and graph. Equation of Axis of Symmetry: _____________________ 2. Find the order pair of the vertex and graph. Order Pair of Vertex ______________ 3. Identify the y-‐intercept. State and graph the ordered pair. __________ Graph the point and its point of reflection across the axis of symmetry. Symmetry Point ________________ 4. Identify the x-‐intercepts. Set the equations = to zero and find the x-‐intercepts (or zeros). Use factoring, quadratic formula or completing the square. State and graph the point(s). _____________________________ 5. Evaluate the function for another value of x. Plot the point and its point of reflection across the axis of symmetry. Generally, choose points one less or greater than the x value of the vertex and then choose one two less or greater than the x value of the vertex. Make a table to display values plotted
6. Draw a _________________ through the graphed points.
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Examples 3 and 4 3. Graph y = x2 − 2x +1 4. y = −x
2 + 4x + 5 Equation of Axis of Symmetry: x = _________ What is the vertex? ( _________, __________) y-‐intercept: ( 0 , ____ ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = _______________ Where does the min or max occur? x = __________ What are the domain and range? Domain: ___________________ Range : ___________________ x-‐intercepts ____________ and _______________ Find 2 points around the vertex and another 2 points using symmetry.
Equation of Axis of Symmetry: x = _________ What is the vertex? ( _________, __________) y-‐intercept: ( 0 , ____ ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = _______________ Where does the min or max occur? x = __________ What are the domain and range? Domain: ___________________ Range : ___________________ x-‐intercepts ____________ and _______________ Find 2 points around the vertex and another 2 points using symmetry.
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Examples 5 and 6 5. Graph y = -‐x2 + 4x -‐ 6 6. y = (1 / 2)x2 + 4x +10 Equation of Axis of Symmetry: x = _________ What is the vertex? ( _________, __________) y-‐intercept: ( 0 , ____ ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = _______________ Where does the min or max occur? x = __________ What are the domain and range? Domain: ___________________ Range : ___________________ x-‐intercepts ____________ and _______________ Find 2 points around the vertex and another 2 points using symmetry.
Equation of Axis of Symmetry: x = _________ What is the vertex? ( _________, __________) y-‐intercept: ( 0 , ____ ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = _______________ Where does the min or max occur? x = __________ What are the domain and range? Domain: ___________________ Range : ___________________ x-‐intercepts ____________ and _______________ Find 2 points around the vertex and another 2 points using symmetry.
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Ex. 7) y = −x2 − 2x + 5 a = _____, b = _____, c = _____ Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________ Ex. 8) y = 1x2x5 2 +− a = _____, b = _____, c = _____ Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________
Ex. 9) y = 23xx
21 2 −+ a = _____, b = _____, c = _____
Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________
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More on Graphing Parabolas 1. y =−x2 +4x−2 Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________ 2. y = 2x2 +4x−1 Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________ 3. y = x2 −4 Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________
4. y = 14x2 − x+3
Opens: ___________ because ________ Axis of Symmetry:____________ Vertex: __________ Min Max of ______ at ________ Domain: Range: X-‐intercepts: __________________ Y-‐intercept: _________
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Graphing with Transformations Date: ____________ After this lesson and practice, I will be able to …
¨ graph quadratic functions in vertex form (using basic transformations). (LT 6) ¨ identify key characteristics of quadratic functions including axis of symmetry, vertex, min/max, y-‐intercept,
domain and range. (LT 7) -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ In addition to graphing from standard form, we can also graph by performing ________________________. This works very well when the quadratic is written in vertex form.
Standard Form 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙+ 𝒄 −
𝒃𝟐𝒂
Vertex Form 𝒚 = 𝒂(𝒙− 𝒉)𝟐 + 𝒌
h
𝑦 = 𝑥! − 4𝑥 + 4 𝑦 = (𝑥 − 2)!
𝑦 = 𝑥! + 6𝑥 + 8 𝑦 = (𝑥 + 3)! − 1
𝑦 = −3𝑥! − 12𝑥 − 8 𝑦 = −3(𝑥 + 2)! + 4
𝑦 = 2𝑥! + 12𝑥 + 19 𝑦 = 2(𝑥 + 3)! + 1
Compare the values of − !!! and h in each row. Write a formula to show the relationship between − !
!! and h.
Vertex Form of a Quadratic Equation: The vertex is _________________________ and the axis of symmetry line is ____________________ Graph the equations on your calculator (Use a window of -‐7 to 7.) Find the vertex of each. y=(x-‐2)2+ 1 ___________ y=(x-‐3)2-‐5 _____________ y=(x+4)2-‐3 ________________ Now let’s transform the parent function y = x2. Go to www.desmos.com and click Start Graphing. In the first box, type the parent function. You can use the carrot button (^) for square. You should see a red parabola appear on the graph. In the second box, write the following equation: 𝑦 = 𝑎𝑥!. Click the blue button that appears to make a slider. Experiment with the slider, then fill in the blanks below … When a is negative, the parabola ______________________________ (this is called a reflection) When 𝑎 > 0, the parabola _____________________________________________
When 0 < 𝑎 < 1, the parabola _____________________________________________
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Add parenthesis and the letter h to Box 2 so that it says this: 𝑦 = 𝑎(𝑥 − ℎ)!.
When ℎ > 0, the parabola shifts _________________________ When ℎ < 0, the parabola shifts _________________________
Now go back to Box 2 and add + k so that Box 1 says this: 𝑦 = 𝑎(𝑥 − ℎ)! + 𝑘.
When 𝑘 > 0, the parabola shifts _________________________
When 𝑘 < 0, the parabola shifts _________________________ Example 1: Describe the transformations applied to the parent quadratic function. Then determine the vertex, equation of the axis of symmetry, and whether the parabola opens up or down (Has a min or max).
a. !!y = (x −4)2 −5 b. !!y = −2(x +3)
2 +6 c. !!y = −12 x
2 −4 Transformation: Equation of AOS: Vertex: Min or Max
Example2: Graph !!y = −2(x −4)2 +3 .
1. Determine how far up/down/right/left, reflections and the vertex
shifts, then graph it.
2. State the equation of the axis of symmetry and graph.
2. Determine if the parabola opens up or down (min or max).
4. When a=1, the points are graphed like this from the vertex: over-‐1-‐up-‐1, over-‐2-‐up-‐4, over-‐3-‐up-‐9. If a is not 1 then we have a vertical stretch of a. Multiply the “up” part of this pattern by a. Then graph three points on either side of the parabola. 5. Connect the points with a smooth curve.
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Example 3: Graph !!y = 12(x +2)
2 −4 Example 4: Graph 𝑦 = −2 𝑥 − 3 ! + 9
Example 5: Graph y =−3(x+4)2 +2 Example 6: Graph 𝑦 = .5 𝑥 + 1 ! − 5
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Example 7: Graph y =(x+2)2 Example 8: Graph 𝑦 = − 𝑥 − 1 ! + 4
Example 9: Graph y =3(x+4)2 −6 Example 10: Graph 𝑦 = !
!𝑥! − 3
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Writing Quadratic Functions Date: ____________ After this lesson and practice, I will be able to …
¨ rewrite quadratic equations from standard to vertex and vice versa. (LT 8) ¨ can write quadratic equations given a graph or given a vertex and the y-intercept (without a calculator)..
(LT 9) ¨ write quadratic expressions/functions/equations given the roots/zeros/x-‐intercepts/solutions. (LT 10)
-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ We have explored several properties of quadratic equations written in _________________ form and _______________ form. Let’s discuss the advantages of each equation form: Advantages to Standard Form Advantages to Vertex Form Rewriting Quadratic Equations (LT 8) Since each equation form has advantages, it is important to be able to rewrite equations in multiple forms. Example 1: Rewrite the vertex form equation in standard form and vice versa. a. !!y =2x
2 +10x +7 b. !!y = −3(x +2)2 −6
c. y = x2 + 2x + 4 d. y = 3x2 −12x +17
e. y = − 12x − 4( )
2+10 f. y = 2 x +5( )
2−30
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Writing Quadratic Equations from a Graph (LT 9) Another important skill is being able to write quadratic equations when given the graph or points. Example 2: Write the equation for each parabola in vertex form. a. b. c. ____________________________ _____________________________ __________________________ Example 3: Write the equation of the parabola with the given vertex, y-‐intercept or point. Write in vertex form then rewrite in standard form. a. vertex at (-‐3, 6) and a y-‐intercept of 33. Vertex form: __________________________ standard form: ___________________________ b. . vertex (-‐2,-‐1) and the graph contains the point (4,11)
Vertex form: __________________________ standard form: ___________________________
c. vertex: (50,1000) and the y-‐int. is -‐500
Vertex form: __________________________ standard form: ___________________________
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We can also write the equation of quadratics from the solutions (or roots or zeros or intercepts!). This should look familiar … we did it at the end of the last unit! Remember from last unit: Write an equation with solutions of -‐7 and 3.
Compare y1 = x2 + 4x − 21 and 0 = x 2 + 4x − 21
Let’s compare some vocabulary:
Term Definition Written as Answered as
Solution Value(s) of x that make the
equations true. These can be real or complex.
x = -‐7 and 3 or {-‐7, 3}
Equations = 0 x2 + 4x – 21 = 0
Does NOT have a vertex, min or
max.
Root
Value(s) of the variable that cause the polynomial to evaluate to zero.
These can be real or complex.
x = -‐7 and 3
Polynomial Expression x2 + 4x – 21
Does NOT have a vertex, min or
max.
Zero
Value(s) in the argument for which
the function is zero. These can be real or complex.
x = -‐7 and 3
Function as f(x) or y= f(x) = x2 + 4x – 21 or y = x2 + 4x – 21
Has a y-‐
intercept, min or max and vertex.
x-‐intercept
An ordered pair on the graph that
has a y value of zero. These must be real.
(-‐7,0) and
(3,0)
Function as f(x) or y= f(x) = x2 + 4x – 21 or y = x2 + 4x – 21
Has a y-‐
intercept, min or max and vertex.
Forms of Quadratics: Expression ax2+bx+c Standard Form ax2+bx+c=0 y=ax2+bx+c x-‐intercept Form y=(x-‐a)(x-‐b) Factored form: y=(x-‐a)(x-‐b) or 0=(x-‐a)(x-‐b) Vertex Form y=a(x-‐h)2+k
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Given The Roots/Zeros/Intercepts/Solutions (LT 10) Example 1: Write a quadratic function in x-‐intercept/factored form with the following zeros: (Since this asks for zeros and y-‐intercepts the equation should be y = or f(x) = ) A) 2 and -‐4 B) 4 and 4 C) 0 and -‐7
_______________________ _____________________ ____________________ Graph the equation from Example 1A in your graphing calculator or on Desmos. What point appears to be the vertex of this equation? _________ How does the x-‐coordinate of the vertex relate to the two x-‐intercepts? Example 2: Write a quadratic equation in x-‐intercept or factored form with the following zeros. Then find the vertex of each equation. (Since this asks for zeros and x-‐intercept and asks you to find vertex the equation should be y = or f(x) =)
A) -‐1 and 3 *B) -‐4i and 4i C) 0 and 8 ________________ _______ _______________ _________ ________________ _______ Example 3: Write a quadratic equation in standard form that has the following solutions. (Since this asks for solutions ax2+bx+c=0. This is really from the another unit.) A. -‐3i and 3i ___________________ B. 0 and 2 ___________________ C. 3 and -‐5 ____________________ Example 4: Write a quadratic equation in standard form that has the following x-‐intercepts. (Since it says x-‐intercepts standard must =y so y=ax2+bx+c) A. (0,0) a,d (-‐3,0) ______________________ B. (5,0) and (-‐2,0) _____________________
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Return of Completing the Square! Date: ____________ After this lesson and practice, I will be able to …
¨ write quadratic equations in vertex form by completing the square. (LT 11) -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ In the last unit, we completed the square as a way to solve quadratic equations. Now we will use the same strategy to rewrite equations into vertex form. Warm Up: Solve this equation by Completing the Square: 0 = x2 −10x + 22 Now Rewrite y = x2 −10x + 22 in vertex form using completing the square That form of quadratic equation should look familiar … what form does it look like? So in addition to being a nice way to solve quadratics, Completing the Square is also a nice way to convert Standard Form equations to Vertex form. Let’s practice!
Write y = x2 +6x+ 2 in vertex form by completing the square, and state the vertex.
How could you verify your answer?
Example 5: Rewrite each equation into vertex form then state the vertex.
a) y = x2 −10x − 2 b) y = x2 +5x+3 c) y = −x2 − 4x+5
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Ifa ≠1, then first divide it out of the x2 and x terms.
d) y = 3x2 −6x+ 4 e) y = −2x2 −16x − 25 f) y = 2x2 +3x+ 2 Example 6: Write each equation in vertex form by completing the square. Identify the vertex.
A) y = 2x2 −8x +1 B) y = x2 + 4x − 7 C) 22 28 99y x x= − +
Example 7: The function P(s) = −s2 +120s− 2000models the monthly profit P from selling sweaters at price s. a) Use completing the square to rewrite the function in vertex form. P(s) = ________________________________ b) Determine the maximum monthly profit, and the price at which the sweaters should be sold to attain that maximum. Sell at $___________ and earn $_____________ max profit. c) Describe the practical domain and range for the profit function. Domain: _____________________ Range: _____________________
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Return of Real World Problems! Date: ____________ After this lesson and practice, I will be able to …
¨ apply quadratics functions to real life situations without using the graphing calculator. (LT 4R) -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ In previous lessons we have used our calculator to answer questions about real life situations. Now lets do this with what we have recently learned. Example 1: The income from ticket sales for a concert is modeled by the function I(p) = -50p2 + 6000p, where p is the price of a ticket. a. Calculate the maximum value of the function. (In other words, how high does
the income go?) __________________ b. What price should be charged in order to attain the maximum income? _________________ Do part a and b again, but this time, use the graph in your calculator. Example 2: A pair of numbers has a sum of 8. Find their maximum product. _________________ Equation: _____________________
Example 3: A rectangle has a perimeter of 24 inches. What dimensions would maximize the area? ___________________ A(x) = ________________________
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Example 4: A rectangle has the dimensions w( ) and 260− 2w( ) in feet. ___________________
What width will maximize the area? What is the maximum area? A(w) = __________________________ ___________________
Example 5: A company knows that –2.5p +500 models the number of unicycles it sells per month, where p is
the price of a unicycle.
Revenue from sales is the _____________________ times the _________________________________.
What price will maximize revenue? _____________ What is the maximum revenue? __________
R(p) = ____________________________
Example 6: An apartment complex has 100 two-‐bedroom units. The monthly profit in dollars realized from renting out the apartments is given by: P(x) = -‐10x2 + 1760x – 50,000 a. How many units must be rented out to maximize profits? ____________________ b. What is the maximum profit realizable? ____________________ c. How many apartments must be rented before a profit is made? ___________________
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Example 7 : A bakery knows from market research that -‐5p + 100 models the of number of pies sold on average day, where p is the price of a pie in dollars. (Hint: make into Revenue by……) a. Write an equation to represent the revenue the bakery can expect on an average day. R(p) = ____________________________
b. What price will maximize the revenue? ____________________ c. What is the maximum revenue? ____________________
Example 8: Profit earned from sponsoring a basketball game between teachers and students is modeled by y = −100x2 +750x − 650 , where x is the price of a ticket. Graph the function on your calculator, make a sketch below that includes the important information, and answer the following questions. a. What is the domain of the function? __________________ b. What profit is earned if the price of the ticket is $5? __________ c. What price will lead to max profit? ___________ What is the max profit? __________
Example 9: Smoke jumpers are firefighters who parachute into areas near forest fires. A jumper’s height during free fall (before the parachute opens) is modeled by: h(t) = −16t 2 + height of plane
where t is time in seconds. If a jumper’s plane is 1400 ft. off the ground when he jumps, how long is he in free fall before his parachute opens at 1000 ft.? h(t) = ___________________________
Example 10 If Jimmy Fallon dropped a T.V. from a height of 1000 feet, how long would it take to hit the ground? How long would it take to land on a building 20 feet tall. Use the formula: h(t) = -‐16t2 + s h(t) = ____________________ h = height when the object landed. If it landed on the ground, then h = ______.
If it landed on the top of a 20 ft. building, then h = ______. t = the time it takes for the object to fall. s = height from which the object is dropped or…. where the object started! _____________ time to ground ______________ time to building ______________
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Unit 2-1 12. I can use the discriminant to determine the number and type of solutions/zeros.
As we’ve seen, quadratic equations can have __________ or _________________ solutions. Let’s discover how to quickly determine which type of solution a given quadratic equation has …
Discriminant: Given , the discriminant is _________________. The discriminant does not include the radical symbol! Fill out the table below.
Equation Value of Discriminant
b2 – 4ac
Number (0, 1 or 2) and Type of Solutions (Real or Complex)
A) 3x2 −5x = 2
B) 2x2 +8x =−12
C) 9x2 +12x + 4 = 0
D)
E) 2x2 =7x−8
Summary:
Value of Discriminant
Number and Types of Solutions
Positive
Zero
Negative
2 0ax bx c+ + =
24 8 1 0x x− + =
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Example 2: Find the discriminant and give the number and type of solutions. Do not solve. A) B) C) 0 real, 2 complex 1 real 2 real
0 real, 2 complex 1 real 2 real
0 real, 2 complex 1 real 2 real
One more use for the discriminant! If the discriminant is a perfect square, the quadratic expression is factorable under rational numbers. If it is not a perfect square, the quadratic expression is prime under rational numbers.
2 10 23 0x x+ + = 2 10 25 0x x+ + = 2 10 27 0x x+ + =
