Quadratics Chapter: Completing the Square Pre...

Unit: Quadratics Chapter: Completing the Square Pre‐Calc 11

Review of Linear Functions The general form of the function is y = mx +b.

Ex.1 – Sketch the following linear functions using the slope/intercept form: a) y = 2x – 3 b) y = –3x + 6 c) y = –2 d) 3x – 6y = 12

Quadratic Functions Ex.1 – Sketch the graph of the following Quadratic functions using your graphing calculator: a) y = x2 b) y = –x2 c) y = x2 – 6 a) y = –x2 + 6 b) y = x2 – 5x – 6 c) y = –x2 – 5x + 6 Notice the location of the y‐intercept in the above graphs.

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Can we accurately sketch a quadratic – without a graphing calculator? Notice that each quadratic has a Vertex – what is the location of this vertex? Complete the square to find the vertex

Completing the Square a) y = x2 + 4x + 6 b) y = x2 + 6x + 8 c) y = x2 + 8x + 12

When a quadratic is written in the vertex form y a(x p)2 q , the vertex is ),( qp

ALSO, Graph the following (Use x‐int, y‐int, and Vertex)

d) y = x2 ‐ 6x + 5 e) y = x2 – 2x – 8

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Worksheet Name:_____________________ Find the vertex (minimum point) of the following quadratic equations by completing he square, then factor to find the x intercepts and y intercept and sketch the graph: a) y = x2 + 8x + 15 b) y = x2 + 10x + 21 c) y = x2 + 2x + 1 d) y = x2 + 2x – 3 e) y = x2 + 4x – 5 f) y = x2 + 12x – 28

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g) y = x2 – 10x – 24 h) y = x2 – 6x – 16 i) y = x2 – 2x – 8 j) y = x2 – 6x + 5 k) y = x2 – 8x + 15 l) y = x2 – 2x + 1

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Unit: Quadratics Chapter: Graphing Vertex Form Pre‐Calc 11

Carefully sketching (graphing) using the vertex: Ex.1: Carefully sketch the graph of the following: a) y = (x – 1)2 – 4 b) y = – (x + 3)2 + 5

c) y = 2(x + 5)2 – 4 d) y = –3(x + 1)2 + 6 Ex.2: Complete the square and carefully sketch the graph of the following: a) y = –x2 – 6x – 7 b) y = 3x2 – 12x + 6

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Worksheet Name:_____________________ 1. Carefully sketch the graph of the following: a) y = (x + 5)2 – 6 b) y = – (x – 2)2 + 4

c) y = –2(x – 4)2 + 5 d) y = 3(x – 1)2 – 5 e) y = 2(x – 1)2 – 1 f) y = –2(x + 3)2 + 3

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2. Complete the square and carefully sketch the graph of the following: a) y = –x2 – 6x – 4 b) y = 2x2 – 20x + 44 c) y = –2x2 + 16x – 27 d) y = 3x2 – 18x + 23

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Unit: Quadratics Chapter: Solving from Vertex Form Pre‐Calc 11

Solving Quadratics using factoring

We know from earlier that we can find the solution to quadratic equations (x2) by setting equal to zero and factor the equation. Once factors each of the factors can be solved (AKA: zeros/roots)

Ex.1: Solve the following quadratic equations by factoring: a) x2 + 6x +5 = 0 b) –x2 + 6x – 8 = 0 c) x2 – 2x – 3 = 0 Ex.2: Complete the square of the quadratic equations above and indicate the vertex: a) b) c)

V( , ) V( , ) V( , ) Ex.3: Care fully sketch the graph of the above quadratics using your the vertex: a) b) c)

Notice where the sketched graph crosses the x‐axis ‐ these are the solutions (AKA: zeros/roots)! Therefore, the vertex form can be used to solve quadratics – we don’t want to graph them....

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Solving Quadratics using the vertex form

Given an a quadratic equation that we can graph y = a(x – p)2 + q it may or may not cross the x‐axis. The x‐axis is where y = 0, so in effect at the x‐axis 0 = a(x – p)2 + q. We can use this fact to solve the equation algebraically – find the zeros(roots). Ex.1: Solve for x using your vertex form equations above (= 0): a) b) c) Ex.2: Find the roots/zeros/solutions of the following equations (by solving for x): a) y = (x – 1)2 – 9 b) y = –(x – 3)2 + 4 c) y = –(x + 2)2 + 9 Ex.3: Find the roots/zeros/solutions of the following equations (by solving for x) first using the vertex form and

then using factoring: a) y = x2 – 4x b) y = –x2 – 10x – 16 c) y = x2 – 2x – 9


Its not going to work out that nice all the time – roots are going to get involved (yes – radicals!) Ex.4: Find the roots/zeros/solutions of the following equations (by solving for x): a) y = x2 – 6x – 3 b) y = 2x2 – 8x + 1 c) y = –3x2 – 12x + 7

Unit: Quadratics Chapter: Solving from Vertex Form Pre‐Calc 11 Worksheet Name:_____________________ 1. Find the roots/zeros/solutions of the following equations (by solving for x): a) y = –(x – 4)2 + 9 b) y = –2(x – 3)2 + 1 c) y = 3(x + 2)2 – 9 2. Find the roots/zeros/solutions of the following equations (by solving for x): a) y = x2 – 10x – 9 b) y = –2x2 – 12x – 10 c) y = 3x2 + 24x + 36 3. Find the roots/zeros/solutions of the following equations (by solving for x): a) y = x2 – 6x + 5 b) y = x2 + 4x – 3 c) y = x2 + 10x + 13

Unit: Quadratics Chapter: Solving from Vertex Form Pre‐Calc 11 4. Find the roots/zeros/solutions of the following equations (by solving for x): a) y = 2x2 – 4x + 3 b) y = 3x2 + 6x – 2 c) y = 5x2 + 10x – 4 d) y = –2x2 – 8x + 5 e) y = –3x2 + 12x – 5 f) y = –4x2 + 24x – 5

Unit: Quadratics Chapter: Solving with Quad Formula Pre‐Calc 11

Quadratic Formula: To solve (roots) a quadratic written in the vertex form: y = ax2+bx+c Lets look at the general solution using our method of solving:

a) y = 3x2 + 12x + 4 b) y = ax2+bx+c

Use the Quadratic Formula to find the roots of: a) 8x2 + 14x – 15 = 0 b) 10x2 = x + 3 c) 4x2 + 9 = 12x

The Discriminant: The discriminant tells us the number of solutions (roots) there are! a) 3x2 + 4x – 2 = 0 b) 3x2 + 4x – 1 = 0 c) 5x2 – 10x – 20 = 0

x b b2 4ac

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acb 42

Unit: Quadratics Chapter: Solving with Quad Formula Pre‐Calc 11

Worksheet Name:_____________________ Use the quadratic formula to solve the following equations – use the discriminant to check the number of roots before you solve: a) y = 2x2 + 7x + 6 b) y = 2x2 + 13x + 15 c) y = 2x2 + 5x + 2 d) y = 4x2 + 4x – 15 e) y = 4x2 – 28x + 49 f) y = 6x2 – 6x – 12 g) y = 15 – 7x – 2x2 h) y = –2x2 + 11x – 12 i) y = –4 + 4x – x2

Unit: Quadratics Chapter: Completing the Square II Pre‐Calc 11

Recall: when a quadratic is written in the vertex form: y a(x p)2 q , the vertex isV (p,q) Nature of the Roots: x‐intercepts = solutions = roots Ex.1: Complete the square to find the vertex, carefully sketch the graph and note the solution/roots: a) y = –x2 + 2x + 3 b) y = x2 + 6x + 9 c) y = x2 + 4x + 6 Ex.2: Solve each of the above. What do you notice about the roots/solutions in each case? We have (almost) everything we need to graph and solve every quadratic – but there are hard ones!

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Ex.3: Complete the square to find the vertex, solve (or factor) to find the intercepts, Indicate the nature of the roots, note the y‐intercept and draw a sketch of the graph. Verify with the calculator. a) 3x2 + 9x + 4 b) 6x2 – 7x – 3 c) 4x2 – 12x + 9

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Worksheet Name:_____________________ Complete the square to find the vertex, factor to find the intercepts, note the y‐intercept and draw a sketch of the graph. Verify your sketch using the graphing calculator. a) y = 2x2 + 10x + 5 b) y = 2x2 + 16x + 15 c) y = 2x2 + 8x + 5 d) y = 4x2 + 4x – 15 e) y = 4x2 + 16x – 3 f) y = 6x2 –6x – 10

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Unit: Quadratics Chapter: Word Problems Pre‐Calc 11

Maximum/Minimum & other Word Problems: When a quadratic is written in the vertex form: y = ax2+bx+c = 0 In the “real world” engineers are asked too design toward maximizing or minimizing. When a problem can be solved with a quadratic, it will have either a maximum or minimum. Ex.1: Two numbers have a difference of 10. Their product is a minimum. What are the numbers?

Ex.2: A rectangular lot is bounded on one side by a river and on the other three sides by a total of 80m of fencing. Determine the dimensions of the largest possible (area) lot.

Ex.3: An App is being marketed to people at a starting price of $20. Three hundred people are willing to buy

the app at that price. Through market research the company finds that for every $5 increase in price, there are 30 fewer people willing to buy the App. At what price could the app be sold for to product the maximum profit?

Ex.4: An open‐topped box is being made from a piece of cardboard measuring 16 cm by 24 cm. The sides of the box are formed when four congruent squares are cut from the corners, as shown. The base of the box is to have an area of 250 cm2.

a) What is the side length (x) of the squares to be cur out? b) What are the dimensions of the box?

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Worksheet Name:_____________________ When a quadratic is written in the vertex form: y = ax2+bx+c = 0

x b b2 4ac

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Other Word Problems:

BARN

Unit: Quadratics Review Pre‐Calc 11 Calculator Permitted ‐ SHOW YOUR WORK Name:________________________ Date: ________

Part 1: Understanding Quadratics 1. By inspection indicate whether the following quadratic equations when graphed will open up or open down and give the y‐intercept: a) y = 10 – 2x – x2 b) y = 5x – 4x2 – 2 c) y = 7 – 16x + 2x2 open: y = open: y = open: y = 2. Complete the square to find the vertex (minimum point) of the following quadratic equations, then factor to

find the x intercepts and y intercept and make a rough sketch the graph including the intercepts: a) y = x2 – 2x – 63 b) y = x2 – 4x – 32 c) y = 2x2 – 16x + 30 V = V = V = x = x = x = y = y = y =

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Unit: Quadratics Review Pre‐Calc 11 Part 2: Graphing Quadratics 3. Use the vertex form to carefully graph the following quadratic equations. a) y = (x – 1)2 – 6 b) y = –(x + 3)2 + 6 c) y = –2(x – 3)2 + 5 4. Complete the square to find the vertex (minimum point) of the following quadratic equations by completing

the square, then use the vertex and y intercept to sketch the graph: a) y = 2x2 + 12x + 13 b) y = 3x2 – 12x + 6 c) y = 5 + 15x – 3x2 V = V = V = y = y = y =

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Unit: Quadratics Review Pre‐Calc 11 Part 3: Solving Quadratics 5. Find the x‐intercepts by solving your completed square from above – leave your answer as a mixed radical

in lowest terms: a) b) c) x = x = x = 6. Use the discriminant (∆ = b2 – 4ac) to show whether the following equations have 2 roots, 1 root or No roots: a) 3x2 – 2x + 5 = 0 b) 2x2 + 2x – 1 = 0 c) 4x2 – 2x – 1 = 0 7. Use the quadratic equation to solve the following equations: (Show your work ‐ Leave in mixed radical form in lowest terms) a) 0 = x2 – 8x – 20 b) 6x2 + 11x = 10 c) 12x2 + 33x = 9 d) 20x = 4x2 + 25 e) 2x2 = 4x + 1 f) 4x2 + 1 = 4x

x b b2 4ac

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Unit: Quadratics Review Pre‐Calc 11 Part 4: Solving Quadratics 8. A rectangular pen is to be constructed against the wall of a barn using 120m of fence (the wall is used as

one side of the pen). a) What are the sides and length of the pen that give an area

inside the pen of approximately 1200m2? b) What are the sides and length of the pen that give an area

inside the pen of approximately 1500m2? c) What are the sides and length of the pen that give a

maximum area inside the pen? What is the maximum area?

9. A rectangular trough is to be made from a strip of metal 1m wide. The metal is to be bent at right angles so it as to produce a trough with the maximum cross‐sectional area. How high are the sides of the trough?

10. A young entrepreneur decides to create infinity scarves to sell to raise money for a mathematics field trip.

She does a market survey and learns that 100 people would be willing to buy the scarves for $20. She also finds out that for every $2 increase in the purchase price of the scarf, 5 less people will buy the scarf.

a) How many scarves must be sold to bring in $1760? b) How many scarves must be sold to make the

maximum profit? What is the maximum profit?

Unit: Quadratics Review Pre‐Calc 11 11. Two numbers have a difference of 24. Find the 12. The sum of two numbers is 50. The sum of numbers if the sum of their squares is a maximum. their squares is a minimum. What are the What are the numbers? numbers? 13. A mural is being painted on an outside wall that is 16m wide and 10m tall. A boarder of a uniform width surrounds the mural. The mural is to cover only 80% of the area of the wall.

a) What are the dimensions of the mural? b) How wide is the boarder? 14. An open‐topped box is being made from a piece of cardboard measuring 50 cm by 75 cm. The sides of the

box are formed when four congruent squares are cut from the corners, as shown.. a) If the base of the box is to have an area of 2000cm2

i) What is the side length of the squares to be cut out? ii) What are the dimensions of the box? b) If the base of the box is to have an area of 3000cm2

i) What is the side length of the squares to be cut out? ii) What are the dimensions of the box? 15. A car travelling at a speed of v km/h needs a stopping distance of d metres to stop without skidding. This relationship can be modeled by the function d(v) = 0.007v2 +0.15v. What is the top speed that a car can be travelling at and still stop without skidding:

a) in 30m b) in 70m

Quadratics Chapter: Completing the Square Pre...

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