Quadratic Relations Course Pack PDF

Unit 3: Quadratic Relations Course Pack  Linear vs. Non-Linear Relations Sometimes a curve of best fit is a more appropriate model for data than a line of best fit. This is true when the data points seem to fit a recognizable pattern that is not a straight line. In such a...

Unit 3: Quadratic Relations Course Pack  Linear vs. Non-Linear Relations Sometimes a curve of best fit is a more appropriate model for data than a line of best fit. This is true when the data points seem to fit a recognizable pattern that is not a straight line. In such a case, try to draw a smooth curve that passes through as many of the data points as possible. Example: Complete the following table for the relation y = 2x + 1: x y 1st Diff. 0 1 2 3 4 Conclusion: For constant increments of the independent variable, a relation is __________ if the _______________________ of the dependent variable ____________________. Example: Complete the following table for the relation y = x2 + 2: x y 1st Diff. 2nd Diff. 0 2 4 6 8 Conclusion: For constant increments of the independent variable, a relation is ________________if the _____________________of the dependent variable ________________. A linear relation models a situation where the rate of change is constant. A nonlinear relation models a situation with a variable rate of change. The degree of a polynomial with one-variable is the highest exponent that appears in any term of the expanded form of the polynomial. A one-variable polynomial of degree 2 models a quadratic relation. 2𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑣𝑎𝑙𝑢𝑒 The quadratic coefficient can be found using the formula 𝑎 = 2 301  The Parabola Graph y = x2 on the grid provided. a) Method ① table of values Equation y = x2 The Table of Values x y = x2 -4 -3 -2 -1 0 1 2 3 4 b) Method ② step pattern Fill in the following information about the parabola: What is the What is the What's the "step pattern" of the Over 1 Up___ vertex (the direction of parabola? (How do you move from lowest point on the point to point, starting from the Over 1 Up ___ the graph)? opening? vertex? And it doesn't matter if you go to the right or left) Over 1 Up ___ Since all parabolas have their "over" steps the same, we usually refer to these three numbers as the Step Pattern of the parabola. So, the Step Pattern of this parabola is _____________________. If graphing y = a  x - h +k and the value of "a" ≠ 1 , ______________________ 2 ________________________________________________________________________. 302 Key Terms: Vertex: The highest or lowest point on the quadratic relation. General coordinates: ___________ Axis of Symmetry: a line that splits the relation into two equal halves. General form: ___________ Zeros: the values of x where the relation intersects the x-axis. Y-intercept: the point at which the relation intersects the y-axis. Optimal value: the highest or the lowest y-value the quadratic relation can take. This can be also known as the maximum or minimum value. General form: _______________ Domain: all possible values of the independent variable, x. For example: _______________________ Range: all possible values of the dependent variable, y. For example: ______________________________ Example 1: On the graph below, label the following: a) axis of symmetry b) zeros c) vertex d) y - intercept e) optimal value f) State the domain: ________________ g) State the range: __________________________ 303 Example 2: For the following quadratic relations, fill in the table below. Graph Vertex Optimal Value Maximum/minimum Axis m of Symmetry Zeroes Direction Opening y – intercept Example 3: Using a table of values, graph the quadratic relation 𝑦 = −𝑥 2 + 4𝑥 + 5 on the grid below and complete the blanks. x y Vertex: __________ -2 Optimal value: _______ -1 0 Axis of symmetry: _______ 1 Zeros: ____________ 2 3 y-intercept: _____________ 4 Direction of opening: _________ 5 6 304  Warm-up: Quadratic Relations, Degree of a function, Key Properties of a Parabola 1. Determine the degree of each function and determine which function has a graph that is a parabola. a) y = 2x – 4 _____________________________________ b) y = x(x -1) _____________________________________ c) y = x - 2x + 1 3 2 _____________________________________ 2. State the y-intercept of the graph and state whether the graph opens upward or downward. a) y= x2 -4x+ 5 ______________________________________ b) y = -x(x +4) ______________________________________ 3. Determine if the relation is quadratic and whether the graph opens upward or downward. a) b) x y x -2 -1 0 1 2 1 0 y 7 6 5 4 3 2 3 3 8 4 15 4. Graph y = x2 - 4x using a table of values to determine: a) the equation of the axis of symmetry _______________________ b) the coordinates of the vertex _______________________ c) the y-intercept _______________________ d) the zeros _______________________ e) the optimal value _______________________ x y -5 -4 -3 -2 -1 0 1 2 3 4 5 305  The Domain and Range of A Function Relation: A relation is a set of ordered pairs. Function: is a relation in which there is one and only one dependent value (output) for each independent value (input). An alternative definition: A function is a relation in which no two ordered pairs have the same first coordinate. Ways of Representing Functions 1. TABLE OF VALUES 2. COORDINATES in a SET x y f = {(2,1), (0,1), (3,1), (4,1), (7,1)} 2 5 5 7 6 5 3. GRAPH 4. MAPPING (Bubble Diagrams) Domain Range -2 0 4 1 5 8 5 5. a) EQUATION b) Function Notation y = x2 – 5 f(x) = x2 - 5 Vertical Line Test On a graph, the idea of single valued means that no vertical line ever crosses more than one value. If it crosses more than once it is still a valid curve, but is not a function. Domain: The domain is the set of all the independent values or inputs (for example: 𝑥). Range: The range is the set of all the dependent values or outputs (for example: 𝑦). 306 Examples: In each case below, 𝑓 represents a relation. State the domain and range of 𝑓. a) f  {(2, 4), (1,1), (0, 0), (1,1), (2, 4)} D   __________________  R   __________________  b) c) D   __________________  D   __________________  R   __________________  R   __________________  d) e) D   __________________  D   __________________  R   __________________  R   __________________  307 f) g) D   __________________  D   __________________  R   __________________  R   __________________  h) i) D   __________________  D   __________________  R   __________________  R   __________________  j) k) D   __________________  D   __________________  R   __________________  R   __________________  308  Warm Up: Domain and Range 1. State the domain and range for the relations with the following graphs: a) b) c) Domain: __________________ Domain: __________________ Domain:_________________ Range: ___________________ Range: ___________________ Range___________________ 2. State the domain and range for the relations with the following equations: a) 𝑦 = 2𝑥 − 4 b) (𝑥 + 1)2 + (𝑦 − 3)2 = 25 Domain: _________________________ Domain: _________________________ Range: __________________________ Range: __________________________ c) 𝑦 = √𝑥 Domain: ________________________ Range: __________________________ 309  Investigating Transformations: Graphing 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 What is the effect of varying ‘𝒌’? INSTRUCTIONS:  Use graphing software to graph the pairs of quadratic relations below  Sketch and label each graph using a different color  State the vertex of each parabola beside its equation  Compare the graphs in each pair 1) y  x2 2) y  x 2 y  x2  3 y  x2  6 Compare: Compare: CONCLUSION:  When we graph the quadratic relation, y  x  k , the vertex of the parabola has coordinates 2 ___________.  When ‘𝑘’ is positive, the graph of y  x is _________________ translated _________ ‘|𝑘|’ units. 2  When ‘𝑘’ is negative, the graph of y  x is _________________ translated __________ ‘|𝑘|’ units 2 APPLY: State the equations of the graphs below: _________________ __________________ _________________ y y y 10 4 2 8 x 2 6 -4 -2 2 4 4 x -2 -4 -2 2 4 2 x -4 -4 -2 2 4 -2 GRAPH: Without graphing software, sketch the graphs on the same axes to y the right. State the vertex of each graph. 2 x a) y  x  1 2 -4 -2 2 4 -2 b) y  x  5 2 -4 310  Investigating Transformations: Graphing 𝑦 = 𝑎(𝑥– ℎ)2 + 𝑘 What is the effect of varying ‘𝒉’? INSTRUCTIONS:  Use graphing software to graph the pairs of quadratic relations below  Sketch and label each graph using a different color  State the vertex of each parabola beside its equation  Compare the graphs in each pair 1) y  x2 2) y  x 2 y   x  5 y  x  3 2 2 Compare: Compare: CONCLUSION: When we graph the quadratic relation, y  x  h  , the vertex of the parabola has 2  coordinates ___________. When ‘h’ is positive, the graph of y  x  h  is ________________ translated to the __________ 2  ‘|ℎ|’ units. When ‘h’ is negative, the graph of y  x  h  is ________________ translated to the __________ 2  ‘|ℎ|’ units. APPLY: State the equations of the graphs below: __________________ ___________________ __________________ y y y 2 2 2 x x x 2 4 6 -6 -4 -2 2 4 GRAPH: Without graphing software, sketch the graphs on the same axes to the right. State the vertex of each graph. y a) y  x  4 2 4 b) y  x  6 2 2 x -6 -4 -2 2 4 311  Graphing Quadratic Relations With Translations Graph the following quadratic relations. List the transformations to the graph of y = x 2 that you applied. State the coordinates of the vertex, the axis of symmetry, x-intercept(s), y- intercept, the domain and the range. a) y = x 2 - 4 b) y = (x + 4)2 + 2 Transformations: Transformations: _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ Vertex =________________ Vertex =________________ Axis of Symmetry: ________________ Axis of Symmetry: _______________ x-intercepts: ________________ x-intercepts: ________________ Domain: ________________ Domain: ________________ Range: ________________ Range: ________________ 312  Graphing Quadratic Functions by Hand Without using the graphing calculator, graph the following quadratic functions. List the transformations to the graph of y = x2 that you made. State the coordinates of the vertex, the axis of symmetry and any x-intercepts. a) y = x 2 - 7 b) y = (x + 5)2 + 4 Vertex =__________ x-intercepts:___________ Vertex =__________ x-intercepts:___________ Axis of Symmetry: ________________ Axis of Symmetry: ________________ Transformations: Transformations: __________________________________________ __________________________________________ __________________________________________ __________________________________________ __________________________________________ __________________________________________ 2. Given the following graphs, determine the equation of the quadratic functions. 313  Investigating Transformations: Graphing 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 What is the effect of varying ‘a’? INSTRUCTIONS:  Use graphing software to graph the pairs of quadratic relations below  Sketch and label each graph using a different color  Compare the graphs in each pair 1) y  x2 2) y  x 2 y  x2 y  2x 2 Compare: Compare: 3) y  x 2 4) y  x 2 1 2 y x y  3x 2 3 Compare: Compare: CONCLUSION:  When ‘𝑎’ is positive, the graph of y  ax2 opens _________________.  When ‘𝑎’ is negative, the graph of y  ax2 is reflected in the ___________ and opens _________________.  When we graph the quadratic relation, y  ax2 , the ‘𝑎’ has the effect of compressing or stretching the graph vertically. o When a  1 , the graph is ________________ ______________ by a factor of |𝑎|. o When 0  a  1 , the graph is ________________ ______________ by a factor of |𝑎|. 314  Transformations of Quadratic Relations Summary -If |𝑎| > 1  vertical expansion by a factor of |𝑎| -If 0 < |𝑎| < 1  vertical compression by a factor of |𝑎| -If 𝑎 < 0  reflection in the x-axis 2 y = a (x - h) + k If ℎ > 0, If 𝑘 > 0,  horizontal translation |ℎ|  vertical translation |𝑘| units to the right units up If ℎ < 0, If 𝑘 < 0,  horizontal translation |ℎ| units to  vertical translation |𝑘| the left units down 315  Graphing Transformations Graph the following quadratic relations. a) 𝑦 = 2(𝑥 − 4)2 − 6 b) y  ( x  2) 2  2 1 c) y  2( x  1) 2  2 d) y   ( x  3) 2  5 2 316  Graphing 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 Putting it All Together! Without using of graphing software, graph the following quadratic functions. List the transformations to the graph of y = x2 that you made. State the coordinates of the vertex, the axis of symmetry, and any intercepts. y = 2  x + 5 + 4 y = -4  x + 3  2 2 a) b) Transformations: Transformations: vertex vertex x-intercepts x-intercepts Axis of Symmetry Axis of Symmetry y-intercept y-intercept 317  1 2 1  x - 5   10 2 c) y= x -8 d) y = - 4 2 Transformations: Transformations: vertex vertex x-intercepts x-intercepts Axis of Symmetry Axis of Symmetry y-intercept y-intercept 318  Graphing Quadratic Relations 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 Graph the following quadratic relations. List the transformations to the graph of y = x2 that was applied. State the coordinates of the vertex, the axis of symmetry, x-intercept(s), y-intercept, the domain and the range. a) 𝑦 = (𝑥 − 1)2 – 4 b) 𝑦 = −(𝑥 + 5)2 Transformations: Transformations: Vertex = _____________ Vertex = _____________ Axis of Symmetry________________ Axis of Symmetry________________ x-intercepts _____________________ x-intercepts _____________________ Domain _____________________________ Domain _____________________________ Range ______________________________ Range ______________________________ 319  1 c) 𝑦 = − 2 (𝑥 − 1)2 + 2 d) 𝑦 = −4(𝑥 − 3)2 + 4 Transformations: Transformations: Vertex = _____________ Vertex = _____________ Axis of Symmetry________________ Axis of Symmetry________________ x-intercepts _____________________ x-intercepts _____________________ Domain _____________________________ Domain _____________________________ Range ______________________________ Range ______________________________ 320  Quiz: Graphing Quadratics in the Form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 Without using graphing software, graph the following quadratic functions. List the transformations to the graph of y = x2 that you made. Fill in the blanks below each graph. 1  x + 3 - 2 2 a) y= b) y = -2x 2 + 8 2 Transformations: Transformations: vertex vertex Zeros Zeros Axis of Symmetry Axis of Symmetry Domain Domain Range Range 321  The Factored Form of a Quadratic Function A quadratic relation in the form y  a( x  r )( x  s) , is said to be in factored form. The zeros are 𝑥 =______ and 𝑥 =______. The axis of symmetry can be determined by using the formula _____________. The axis of symmetry is also the _____________________ of the vertex. A quadratic function can be written in 3 different forms. Key information can be read or found from each particular form. Form General Function Key Information Axis of symmetry standard y  a x 2  bx  c vertex y  a ( x  h)2  k factored y  a ( x  r )( x  s ) 1 Example 1. For the quadratic relation y =  x - 5  x + 3  : 2 a) state the zeros b) determine the axis of symmetry c) determine the vertex d) determine the y-intercept e) Sketch the graph 322 Example 2. Sketch the graph of y  ( x  1)( x  6). Example 3. Determine the equation of the quadratic relation, in factored form, which has: a) zeros of -2 and 4 and a y-intercept of -3. b) x-intercepts of 2 and 7 and a minimum value of -5. 323  Writing Equations of Parabolas Determine the equation of the quadratic relations for each of the following: 1. In vertex form with a vertex of (-2,3) 2. In standard form with zeros -3 and 1 and passing through the point (-1,6). and passing through the point (-1,-7). 3. In standard form with a vertex of (-4, 1) 4. In vertex form with a maximum at (4,-2) and y intercept of -5. and congruent to y = 2x2. 324  Writing Quadratic Relations from Graph Determine the equation of the following quadratic functions. a) b) Factored Form: Factored Form: Standard Form: Standard Form: vertex Form: vertex Form: c) d) Factored Form: Factored Form: Standard Form: Standard Form: vertex Form: vertex Form: 325 e) f) Factored Form: Factored Form: Standard Form: Standard Form: vertex Form: vertex Form: g) h) Factored Form: Factored Form: Standard Form: Standard Form: vertex Form: vertex Form: 326  Using a Formula to Determine the Vertex of a Quadratic Relation A formula can be derived to find the vertex of a quadratic relation given in standard form, 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Complete the square on the standard form of a quadratic relation to derive the formula. Steps Equation: 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 Remove the quadratic coefficient as a common factor from the first TWO terms. Add and subtract the square of half the linear coefficient inside the brackets. Remove the last term from inside the brackets and combine with the constant term. Don’t forget to multiply by the quadratic coefficient! Factor the expression in the brackets as a perfect square trinomial. The General Formula to Determine the Vertex In general, the x-coordinate of the vertex of a quadratic relation, 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0, can be found by using the formula ___________________. The y- coordinate can then be found by substituting the x-coordinate into the original equation. This method will give you the coordinates of the vertex. Example 1. Determine the vertex of the following quadratic relations: 1 3 a) y  2 x 2  4 x  5 b) 𝑦 = 2 𝑥 2 + 𝑥 − 2 c) 𝑦 = 3𝑥 2 + 2𝑥 + 15 327 Example 2. A ball is thrown from the top of a building. Its height, ℎ, in metres, after 𝑡 seconds is ℎ = −5𝑡 2 + 8𝑡 + 10. Determine the maximum height the ball and when it occurs. Example 3. The profit on the school drama production is modelled by the quadratic function, 𝑃 = −60𝑥 2 + 790𝑥 − 1000, where 𝑃 is the profit in dollars and 𝑥 is the price of the ticket, also in dollars. a) Determine the break-even price (the price where there is no profit or loss) for the tickets. b) At what price should the drama department set the tickets to maximize their profit? What is the maximum profit? 328  Vertex Form of a Quadratic Relation Practice 1. Write in the form y = a(x -h)2 + k: Answers: a) y = x2 -6x+8 b) y = x2 + 10x + 14 c) y = 2x2 + 4x + 7 d) y = -2x2 + 4x -I- 5 e) y = 3x2 - 24x + 40 f) y = -5x2 - 20x – 30 2. Sketch the graphs of the parabolas in Exercise 1. 3. Sketch each parabola showing: i) coordinates of the vertex; ii) equation of axis of symmetry; iii) coordinates of two other points on the graph. (-2, 9) (-4, 7) a) y = x2- 6x + 10 b) y=2x2+ 8x+ 7 (0, -5), (-3, -5) c) y = -x2 + 10x – 13 d) r = 3t2 -6t + 8 e) m = -4n - 24n – 20 2 f) m = - 2v2- 16v – 35 4. Sketch each parabola showing: i) coordinates of the vertex; ii) equation of axis of symmetry; iii) coordinates of two other points on the graph. a) y = -½x2 - 2x + 7 b) r = 4t2 + 12t - 5 c) k =-2j2 + 14j - 12 d) y = 3x2 - 4x- 6 e) u = -4v2 + 10v – 7 12. Find the equation of the parabola: a) with vertex (-2, 0), y-intercept 4; b) with vertex (5, 0), y-intercept 25; c) with x-intercept -6, y-intercept 36, axis of symmetry x + 6 = 0. 13. Find the equation of the parabola with vertex (0, 0) which passes through: a) (2, 16) b) (3, -18) c) (2, 6) d) (-3, 15) 2 2 5 1 1 e) (3 ,− 3) f) (4,12) g( ,5) 2 h)( 2 ,− 2) 14. Write the equation of the parabola: a) with vertex (-3, 4), y-intercept -5; b) with vertex (2, -2), x-intercepts 1 and 3; c) with vertex (4, -4), that opens up, and is congruent to y = ½ x2. 15. Sketch each parabola showing: i) coordinates of the vertex; ii) equation of axis of symmetry; iii)coordinates of two other points on the graph. a) y = x2 - 6x + 5 b) w = 2z2 - 8z- 5 c) v = ½t2+ 10t+21 d) p = -3q2+ 18q -20 329  Partial Factoring Recall: If 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 can be factored into 𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠), then this equation can be used to determine the vertex form of the quadratic relation. What if the relation cannot be factored? Another strategy is Partial Factoring. PARTIAL FACTORING: If 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 cannot be factored use the following steps to determine the vertex form of the quadratic relation: 1. Express the relation in the partially factored form 𝑦 = 𝑥(𝑎𝑥 + 𝑏) + 𝑐 2. Set 𝑦 = 𝑐 or 𝑥(𝑎𝑥 + 𝑏) = 0 and solve for 𝑥 to determine two points on the parabola that are the same distance from the axis of symmetry. Both of these points will have a 𝑦-coordinate of 𝑐. 3. Calculate the axis of symmetry, 𝑥 = ℎ, by calculating the mean of the 𝑥-coordinates. 4. Substitute the ℎ-value into 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 to determine the 𝑦-coordinate of the vertex. This gives you 𝑘. 5. Substitute the values of 𝑎, ℎ and 𝑘 into 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘. Example 1. Determine the maximum value of the quadratic relation y  3x2  12 x  29, using partial factoring and express the relation in vertex form. Example 2. Use partial factoring to determine the vertex of the relation y  5x2  40 x  2, and sketch a graph. 330  Quadratic Regression Use a graphing calculator for the following questions. 1. A farming cooperative collected data showing the effect of different amounts of fertilizer, x, in hundreds of kilograms per hectare (kg/ha), on the yield of carrots, y in tons (t). Fertilizer, x (kg/ha) 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Yield, y (t) 0.16 0.46 0.63 0.91 0.96 1.08 1.05 0.88 0.78 a) Enter the data into a graphing calculator and use a quadratic regression to estimate y as a function of x. b) Determine the yield of carrots when 0.8 kg of fertilizer is used. c) Determine the amount of fertilizer used when 0.5 tons of carrots was produced. d) How much fertilizer would you recommend the farmers use? Explain. 2. This table shows the number of imported cars that were sold in Newfoundland between 2003 and 2007. Year 2003 2004 2005 2006 2007 Sales of Imported 3996 3906 3762 3788 4151 Cars a) Enter the data into a graphing calculator and create a scatter plot. b) Use a quadratic regression to determine an algebraic equation to model this relation. c) Use your model to predict how many imported cars were sold in 2008. d) What does your model predict for 2006? Is this prediction accurate? Explain. Answers: 1. a) 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑎 = −0.5328138528, 𝑏 = 1.382294372, 𝑐 = 0.1403030303 b) 0.905 c) 0.293, 2.30 d) 1.30 2. b) 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎 = 76.85714, 𝑏 = −308177.94285714, 𝑐 = 308932906.6 c) 4516 d) 3862 331  Quadratic Inequalities Quadratic inequalities can be solved graphically and algebraically. Since we know how to graph quadratic relations, we can solve quadratic inequalities graphically. Ex. 1. Solve the quadratic inequality 𝒙𝟐 − 𝟒 ≤ 𝟎. A) Graphing By Hand Step 1. Sketch the graph of the Step 2. Look at the graph to determine the quadratic relation, 𝑦 = 𝑥 2 − 4. Plot interval(s) where the relation will be equal to accurate x-intercept(s), if possible. or less than 0 (i.e. the interval(s) where the graph is on or below the x-axis). This will be the solution to the inequality. Step 3. State your solution in the required form. Set notation: ___________________ Interval notation: ________________ Number Line: B) Using an interval chart Step 1. Collect all terms to one side. Step 3. Consider the intervals where the Fully factor the inequality. factor will be positive or negative. Step 4. Determine the overall product of the factors at the bottom of the chart. Step 2. List the factors out on the left of Step 5. State the solution in the required the number lines. form. 332 Ex. 2. Solve the following inequalities by graphing. a) −2(𝑥 − 4)2 + 8 < 0 b ) 0 ≥ 𝑥 2 + 2𝑥 − 8 Ex. 3. Solve the following inequalities using an interval chart. a) 2𝑥 2 + 4𝑥 ≥ 𝑥 2 − 𝑥 − 6 b) – 𝑥 2 + 6𝑥 − 9 < 0 333  Solving Quadratic Inequalities Worksheet 1. Solve the following quadratic inequalities graphically and state the solution in set notation. a) x 2  x  20  0 b)  x 2  3x  54  0 c) x 2  5 x  13  0 d )2 x 2  4 x  30  0 2. Solve the following quadratic inequalities using an interval chart and state the solution in interval notation. a)  3 x 2  6 x  9  0 b) 2 x2  9 x  5  0 c) 9 x2  31x  12 d ) 5  2 x2  3x 5 ) 2 334  Quadratics Warm-Up!!! 1. Without using the graphing calculator graph the following quadratic functions. List the transformations to the graph of y = x2 that you made. State the coordinates of the vertex for each graph. 1 a) y = -2  x + 4  + 9 2 b) y = x 2 - 8 4 Transformations: Transformations: ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ______ ______ 2. Given a quadratic relation, 𝑦 = −2(𝑥 + 3)2 − 3, a) Does the point (-2,-4) lie on the parabola? b) Find two points that lie on the parabola. 335 3. Determine the vertex form of the quadratic function that satisfies each of the following conditions: a) Vertex at (0, 3); opens downward; the same shape as y = x2. b) Vertex at (-2, 5); opens upward; narrower than y = x2. c) Vertex at (1, -4); opens downward; wider than y = x2. d) The graph of y = x2 is expanded vertically by a factor of 3, then translated 5 units right. 1 e) The graph of y =x2 is compressed vertically by a factor of , then translated 5 units 2 down. f) The graph of y = x2 is reflected about the x-axis, expanded vertically by a factor of 3, then translated 2 units left and 7 units up. g) Has a vertical expansion factor of 3 and a vertex of (-2, 6). h) Vertex of (-4, -2) and passes through (-2, -6). i) Vertex (-1,3) that passes through (-2,-5). 336  Maximum and Minimum Word Problems 1. Understand the question and identify variables. 2. Draw a well labeled diagram. 3. Construct an equation, Q, for the quantity to be maximized/minimized. 4. Express Q in terms of one variable only (if there are 2 variables, look for information to create a second equation used to reduce variables). 5. Express the equation in vertex form to find maximum/minimum Note: Profit = Revenue – Cost Number Type: Find two numbers whose sum is 32 and whose product is a maximum. 337 Revenue/Profit Type: The cost of a ticket to a hockey area seating 800 people is $3.00. At this price every ticket is sold. A survey indicates that if the price is increased, attendance will fall by 100 for every $0.50 of increase. What ticket price results in the greatest revenue? What is the greatest revenue? Projectile Type: A ball is thrown into the air from an apartment balcony. The height, h metres, of the ball relative to the ground after t seconds is given by the equation: h(t)= -5t2 +20t +25. a) Find the maximum height of the ball above the ground. b) When did the ball reach the maximum height? c) How high is the balcony above the ground? 338 Area Type: A gardener wants to enclose a rectangular vegetable garden with 60 m of fencing. a) What are the dimensions that will maximize the area? b) What is the maximum area? 339  Worksheet # 1: Quadratic Relations Word Problems 1. A ball is kicked from the ground into the air. The height of the ball, h, in metres, after t seconds is modelled by h= 30t -5t2. How long is the ball in the air? [Answer: 6 s] 2. A ball is kicked from the ground into the air. The height of the ball h, in metres, after t seconds is modelled by h = 30t- 5t2. Find the greatest height the ball reaches. [Answer: 45 m] 3. An object is thrown upward with an initial velocity of v m/s from an initial height of c metres. The height reached by the object, in metres, after t seconds can be modelled by h = -5t2 + vt + c. What is the initial velocity of the object if it reaches its maximum height after 4 s? [Answer: 40m/s] 4. The distance a car travels when it skids is modelled by d= 250 + 50t – 4t2, where d is the length of the skid in metres and t is the time in seconds needed for the car to stop. How long does it take the car to stop? [Answer: 6.25 s] 5. A campground charges $20.00 to camp for one night. They average 56 people each night. A recent survey indicated that for every $1.00 decrease in the nightly price, the number of camping sites rented increases by seven. What price will maximize nightly revenue? Show the steps of your solution. [Answer: price : $14; revenue : $1372] 6. A toy rocket is launched with an initial velocity of 180 m/s. The height of the rocket, in metres, can be modeled by h = - 5t2 + 180t where t is the time in seconds the rocket is in the air. How long will the rocket stay above a height of 1000 m? Show the steps of your solution. [Answer: about 22.3 s] 7. A farmer has $5200 to spend on fencing to make a pen along a river. A local company tells the farmer that they can build the pen for $6.50/m. The farmer tells the company that he wants a rectangular pen with the river as one of the sides. The manager from the company suggests a pen in the shape of a right triangle with the hypotenuse along the river. Which shape should the farmer go with if he wants to get the pen with the greatest area? Show the steps of your solution. [Answer: 80 000 m2 for both shapes] 8. The cost per book, C, in dollars, when a school orders yearbooks is modelled by C = 0.000 05n2 - 0.095n + 66.125, where n is the number of books ordered. How many yearbooks does a school have to order to have the least cost per book? What is the least cost per book? Show the steps of your solution. [Answer: 950 yearbooks, $21 per book] 9. When a ball is tossed into the air, the height, h, in metres, of the ball after t seconds is modelled by h = -5t2 + 20t + 1. After how many seconds, correct to two decimal places, does the ball hit the ground? [Answer: 4.05 s] 10. The city bus company carries, on average, 3500 passengers daily. Each passenger pays $2.25 to ride the bus. Market research has shown that for every $0.25 increase in bus fare, the company loses 50 customers. Write an equation, in standard form, for the revenue, M, in dollars, in terms of the number of $0.25 price increases, n. [Answer: M= -12.5n2 + 762.5n + 7875] 340  Applications of Quadratic Relations II: Optimization Example 1: Farmer Josephine wishes to make the largest possible rectangular vegetable garden using 24 m of fencing. The garden is right behind the back of her house, so she only has to fence 3 sides of the garden. Determine the dimensions of the garden that will maximize the area. Example 2: Tom sells T-shirt for $10. At this price, he is able to sell 30 T-shirts per week. Tom noticed that for every $1 decrease in price, he is able to sell 1 more T-shirt. At what price should he be selling his T-shirt in order to maximize his revenue? 341 Example 3: Mila sells CDs at $20 each. At this price she is able to sell 280 CD per day. However, market research shows that for every $0.50 increase in price, her sales drops by 5 units. At what sale price should she set her CDs to be in order to maximize revenue? What is her maximum revenue at this price? 342  Worksheet #2: Quadratic Relations Word Problems 1. A ball is shot into the air. Its height h, in metres, after t seconds is modelled by h = –4.9t2 + 30t + 1.6. How long will it take the ball to reach a height of 35 m? Show the steps of your solution. [Answer: about 1.46 s] 2. When a ball is thrown into the air, its height, h, in metres, after t seconds is modelled by h = - 4.9(t –2)2 + 20. When will this ball hit the ground? Show the steps of your solution and correct your answer to two decimal places. [Answer: 4.02 s] 3. Suppose that while on the moon, one astronaut tossed a wrench to another astronaut. The height of the wrench, h, in metres, is modelled by h = –0.8t2 + 10t + 1.4. When will the wrench hit the ground? Show the steps of your solution. [Answer: after 12.6 s] 4. A rocket club is studying the performance of some of their rockets. A person stands on a tower, 30 m high, and starts a stopwatch when a launched rocket reaches the height of the tower. Other measurements of height reached by the rocket and time taken to reach that height are made. When a quadratic regression is completed, the height of the rocket, h, in metres is modelled by h = – 4.87t2 + 71.41 t + 30.26, where t is the time in seconds after the stopwatch is started. Predict when the rocket reached a height of only 10 m above the ground. Show the steps of your solution. [Answer: t = – 0.28 or t = 14.9] 5. A rocket is fired from the floor of the Grand Canyon. The height of the rocket, h, in metres, above the floor of the canyon after t seconds is modelled by h = –5t2 + 240t. A person sitting at the top of the canyon, 1734 m above its floor, can only see the rocket when it is above that height. For how many seconds will the rocket be visible? Show the steps of your solution. [Answer: 30.28 s] 6. A rectangle is 7 cm longer than it is wide. The diagonal of the rectangle is 4 cm longer than the longest side. Find the dimensions of the rectangle. Show the steps of your solution. [Answer: 13.4 cm × 20.4 cm] 7. Suppose your student council did a survey on ticket prices for the upcoming Christmas formal dance. They found that 480 students would buy tickets if the price were $5 per ticket. For each $0.10 increase in the price of a ticket, the number of students who will attend the dance drops by 8. Predict the maximum revenue the student council can receive from the sale of tickets. Show the steps of your solution. [Answer: $2420] 8. A public rectangular-shaped pool is 15 m by 30 m. Public pools must be surrounded by a cement walkway of uniform width whose area is at least as large as the area of the pool. What is the minimum width of the walkway? Show the steps of your solution. [Answer: at least 4.2 m wide] 9. The sum of two numbers is 37. The sum of their squares is 756.5. Find the two numbers. Show the steps of your solution. [Answer: 12.5 and 24.5] 10. A cylindrical cup is made so that the sum of its diameter and its height is 25 cm. Find the dimensions of the cup so that the surface area of the cup is a maximum. Show the steps of yours solution. [Answer: diameter 50/3 cm height 25/3 cm] 343  Solving Problems Using Quadratic Relations 1. Write the quadratic relation 𝑦 = −2𝑥 2 + 20𝑥 − 41 in vertex form, and sketch the graph. 2. The underside of a concrete railway underpass forms a parabolic arch. The arch is 30 m wide at the base and 10.8 m high in the center. Can a truck that is 5 m wide and 7.5 m tall get through this underpass? 3. The Next Cup coffee shop sells a special blend of coffee for $2.60 per mug. The shop sells about 200 mugs per day. Customer surveys show that for every $0.05 decrease in the price, the shop will sell 10 more mugs per day. a) Determine the maximum daily revenue from coffee sales and the price per mug for this revenue. b) Write an equation in both standard form and vertex form to model this problem. Then sketch the graph. 4. The height, ℎ, of a football in metres, since it was kicked can be modelled by ℎ = −4.9𝑡 2 + 22.54𝑡 + 1.1, where 𝑡 is in seconds. a) What was the height of the football when the punter kicked it? b) Determine the maximum height of the football, correct to one decimal place, and the time when it reached this maximum height. 344  Quadratic Relations Review 1. Complete the table. Direction of Equation of Axis of Function Vertex Opening Symmetry a) y = x2 - 3 b) y = -(x - 4)2 c) y = -3x2 d) 1 y = (x - 3)2 + 1 2 2. For the parabola defined by y = -2(x + 5)2 - 4: a) Does the parabola have a maximum or minimum value? b) What is the value of the maximum or minimum? c) How many 𝑥-intercepts does this parabola have? d) Express the function in standard form. 3. What is the equation of the parabola that has a vertex at (2,4) and a y-intercept of -4? 4. For the parabola y = 3x2 + 18x + 21: a) Convert the function to vertex form (by completing the square). b) State the vertex of the parabola c) Find its y-intercept. d) Determine the zeros of the function. 5. Determine the x-intercept(s) of y = -2(x + 1)2 + 8. 1 6. Sketch the graphs of y = x2 and y = (x - 2)2  3 2 7. Baz Ketball's jump shot follows the path defined by the equation h = -5t2 + 10t + 3, where h is the ball's distance above the floor, in metres, and t is the time in seconds. a) How far above the floor is the ball when it reaches its peak? b) How long does it take to get there? c) If the basket is 3m above the floor, how long has the ball been in the air when Baz scores? 345 346  − 25 [(2, − )] 4 347 348