A. Quadratic Functions · Quadratic Expressions and EquationsQuadratic Relations in Tables and Graphs: Posing and Solving ProblemsA1.1About 70 minutesLearning goalsI can tell from a table whether a relation is quadratic.I can read a parabola's graph to find maximums, minimums, zeros and the x-values that give a certain y-value.I can pose and solve problems about real situations from tables and graphs.Words to knowquadratic relationA relation whose equation can be written y = ax² + bx + c with a ≠ 0. For equally spaced x-values its second differences are constant.parabolaThe U-shaped graph of a quadratic relation.vertexThe highest or lowest point of a parabola: its maximum or minimum.axis of symmetryThe vertical line through the vertex that splits the parabola into mirror-image halves.zerosThe x-values where y = 0: the x-intercepts of the graph.second differencesThe differences between consecutive first differences.A soccer ball, a table and a questionAt a summer practice in Barrie, a coach kicks a soccer ball straight up from a height of 1 m. A phone app records the height every half second (see the table below). The numbers rise, level off and fall again in a perfectly balanced way. Grade 11 starts with this kind of situation: a quadratic relation given as a table or a graph, and real questions about it. How high did it go? When was it 16 m up? When did it land? You can answer all of these before doing any algebra.Time t (s)00.511.522.533.54Height h (m)19.751619.752119.75169.751Height of the ball every half second.Remember from Grade 10In Grade 10 you learned that a relation is quadratic when its second differences are constant (for equally spaced x-values), that its graph is a parabola, and that a parabola has a vertex, an axis of symmetry, zeros (x-intercepts) and a y-intercept. You also found maximum and minimum values in context. This lesson uses those ideas to pose and solve problems from tables and graphs.Step 1: Is it really quadratic?Use the whole-second rows of the table (equal steps of 1 s). Subtract each height from the next one to get the first differences. Then subtract the first differences to get the second differences. If the second differences are all the same (and not zero), the relation is quadratic.th1st diff.2nd diff.0111622131641155−5−15−10−10−10First differences 15, 5, −5, −15 change, but the second differences are all −10. Constant second differences mean the relation is quadratic. The negative sign means the parabola opens down.Step 2: Read the graphPlot the points and join them with a smooth curve (never with straight segments: the ball does not move in straight pieces). The graph lets you answer three kinds of question:"What is h when t = ...?" Go up from the t-axis to the curve, then across to the h-axis."When is h equal to ...?" Draw a horizontal line at that height and read the t-value of every point where it meets the curve. Usually there are two answers: once going up and once coming down.Special points. The vertex gives the maximum (or minimum). The zeros tell you when the quantity is 0 (for example, when the ball lands).0123404812162024t (s)h (m)(0, 1)(1, 16)(2, 21)(3, 16)The ball's height. The dashed line at h = 16 m meets the curve at t = 1 s (going up) and t = 3 s (coming down). The vertex (2, 21) is the maximum height.Worked exampleExample 1: Solving problems from the ball graphUse the table and the graph of the soccer ball.(a) What is the maximum height, and when does it happen? (b) When is the ball 16 m above the ground? (c) About when does the ball land? (d) For how long is the ball at least 16 m high?(a) The highest point (the vertex) is (2, 21): a maximum height of 21 m at t = 2 s. The table is symmetric about t = 2, which confirms the axis of symmetry is t = 2.(b) The horizontal line h = 16 meets the curve twice. From the table, h = 16 at t = 1 s and t = 3 s.(c) Landing means h = 0. At t = 4 the height is still 1 m, and the ball is falling fast, so the curve hits 0 a little after t = 4. Reading the graph, the ball lands at about t = 4.05 s. (In this unit you will learn to get this exactly with the quadratic formula.)(d) Between t = 1 and t = 3 the curve is on or above the dashed line, so the ball is at least 16 m high for 3 − 1 = 2 s.Maximum 21 m at 2 s; 16 m high at 1 s and 3 s; lands at about 4.05 s; at least 16 m high for 2 s.Posing problemsThe curriculum asks you to pose problems as well as solve them. Posing a good problem means asking a question whose answer the graph or table can actually give, and saying what the answer means. Good question starters:What is the greatest (or least) ...?When (or at what value) does ... reach ...?For what values is ... more than ...?How much does ... change between ... and ...?A question like "How fast was the coach's leg moving?" cannot be answered from this table, so it is not a good question for this data.024681012141618202202004006008001000p ($)R ($)(6, 840)(10, 1000)(14, 840)Revenue R = p(200 − 10p) from selling school hoodies at price p. The shaded band shows the prices from $6 to $14 that give revenue of at least $840. The maximum, $1000, is at p = $10.Worked exampleExample 2: Posing and solving a revenue problemA student council finds that if hoodies cost p dollars, they sell (200 − 10p) of them. The revenue table is:p ($): 4, 6, 8, 10, 12, 14, 16 R ($): 640, 840, 960, 1000, 960, 840, 640Pose and solve two problems from this table and the graph above.Check the pattern: the first differences are 200, 120, 40, −40, −120, −200 and the second differences are all −80, so the relation is quadratic.Problem 1 (posed): What price gives the most revenue, and what is that revenue? The table is symmetric about p = 10, so the vertex is (10, 1000). The best price is $10, giving $1000.Problem 2 (posed): For what prices is the revenue at least $840? R = 840 at p = 6 and p = 14, and the curve is above 840 between them. So any price from $6 to $14 works.Interpret: at $10 they sell 200 − 100 = 100 hoodies. Charging more loses so many sales that revenue drops.Best price $10 for $1000 revenue; revenue is at least $840 for prices from $6 to $14.Reading graphs wellAlways state units (s, m, $).A horizontal line usually meets a parabola twice; give both answers unless the context rules one out.Values read from a graph are estimates. Say "about" and round sensibly.Check that an answer makes sense: negative time, negative length or a fraction of a hoodie should make you stop and think.Just for funWhy did the parabola refuse to argue with the straight line?It could already see both sides of the issue: it's perfectly symmetric.Hands-on activityMake your own quadratic data: the falling-ball tableYou need: a phone with a slow-motion or video camera, a measuring tape, a soft ball (a tennis ball or a rolled-up pair of socks), and a wall or door frame. A grown-up or friend can film.Stick small pieces of tape on a wall at 0.5 m, 1.0 m, 1.5 m and 2.0 m above the floor.Toss the ball gently straight up beside the wall (not toward anyone or anything breakable) while someone films.Step through the video frame by frame. Note the time (the video player shows it) and the height (estimate from the tape marks) at about 6 to 8 moments.Make a table and a graph (by hand on grid paper, or with free graphing software such as Desmos).Pose three questions your graph can answer (for example, What was the greatest height? When was the ball 1 m high?) and answer them.What to look for: the points form a parabola that opens down. The rising and falling halves should be close to mirror images. Your two answers for "when was the ball at 1 m?" should be about equally far before and after the time of the maximum height.Your turn1Using the soccer-ball table, what is the height at t = 1.5 s? Name another time with the same height.Show answer19.75 m. By symmetry about t = 2, the height is also 19.75 m at t = 2.5 s.2Find the first and second differences for x: 0, 1, 2, 3, 4 and y: 3, 5, 11, 21, 35. Is the relation quadratic?Show answerFirst differences: 2, 6, 10, 14. Second differences: 4, 4, 4. Constant, so yes, quadratic (opening up, since they are positive).3Find the second differences for x: 0, 1, 2, 3, 4 and y: 2, 5, 8, 11, 14. Is the relation quadratic?Show answerFirst differences are all 3, so second differences are all 0. The relation is linear, not quadratic.4For the hoodie revenue table, what revenue comes from a price of $8? Which other price gives the same revenue?Show answer$960. By symmetry about p = 10, a price of $12 also gives $960.5A parabola's table has the points (1, 7) and (5, 7). What is the equation of its axis of symmetry?Show answerHalfway between x = 1 and x = 5: x = 3.6Pose one question that the soccer-ball graph can answer and one that it cannot.Show answerSample: Can: "How high was the ball after 0.5 s?" (9.75 m). Cannot: "How heavy was the ball?" The graph only shows height and time.7The ball is 9.75 m high at t = 0.5 s. Without the table, when else is it 9.75 m high? Explain.Show answerThe axis of symmetry is t = 2. 0.5 is 1.5 s before 2, so the other time is 1.5 s after: t = 3.5 s.More practice: warm-up1What shape is the graph of a quadratic relation?Show answerA parabola.2Second differences of a relation are all −6. Does the parabola open up or down?Show answerDown (negative second differences).3What is the vertex of the soccer-ball parabola?Show answer(2, 21).4What does the y-intercept (0, 1) mean for the soccer ball?Show answerThe ball started 1 m above the ground at t = 0.5In the hoodie problem, what does the point (14, 840) mean?Show answerA price of $14 gives revenue of $840.More practice: on your own1x: 0, 1, 2, 3, 4, 5 and y: 0, 9, 16, 21, 24, 25. Is it quadratic? Opening up or down? Estimate the vertex.Show answerFirst differences 9, 7, 5, 3, 1; second differences all −2, so quadratic, opening down. The first differences are shrinking toward 0 near x = 5, so the vertex is about (5, 25). (Indeed the next value would be 24 at x = 6.)2A rocket's height is 0 m at t = 0 s and 0 m again at t = 8 s. When does it reach its maximum height? Why?Show answerAt t = 4 s, halfway between the two zeros, because the axis of symmetry passes through the midpoint of the zeros.3Using the hoodie graph, for what prices is the revenue less than $640 (with 0 < p < 20)?Show answerR = 640 at p = 4 and p = 16, and the curve is below 640 outside that interval: 0 < p < 4 or 16 < p < 20.4A diver's height above the water is shown in a table: t (s): 0, 0.5, 1, 1.5; h (m): 3, 4.25, 3, −0.75. Between which two table times does the diver enter the water? Explain.Show answerh changes from positive (3 at 1 s) to negative (−0.75 at 1.5 s), so h = 0 somewhere between t = 1 s and t = 1.5 s. (The −0.75 just shows that the model has gone past the water surface.)5Pose a 'for how long' problem about the soccer ball and solve it.Show answerSample: For how long is the ball at least 19.75 m high? h = 19.75 at t = 1.5 and t = 2.5, so for 1 s.Challenge1A table for a quadratic relation has x: 0, 2, 4, 6 and y: 5, 1, 5, 17. Find the next value, at x = 8, and explain your method.Show answerFirst differences: −4, 4, 12. Second differences: 8, 8. Next first difference = 12 + 8 = 20, so y = 17 + 20 = 37.2A ball's height h (m) is given by a parabola with zeros at t = −0.2 s and t = 3.8 s (only t ≥ 0 makes sense). When is the maximum height reached, and why is t = −0.2 not part of the real situation?Show answerThe axis of symmetry is halfway between the zeros: (−0.2 + 3.8) ÷ 2 = 1.8 s. The zero at t = −0.2 is before the ball was thrown, so it is outside the domain of the situation (t ≥ 0). It is just where the mathematical curve would cross the axis if extended backward.3Two students read the soccer-ball graph to find when h = 20 m. One says about t = 1.55 s, the other about t = 2.45 s. Who is right? Check with h = −5t² + 20t + 1.Show answerBoth are right: there are two times. h(1.55) = −5(2.4025) + 31 + 1 ≈ 19.99 and h(2.45) = −5(6.0025) + 49 + 1 ≈ 19.99, so both are about 20 m. They are symmetric about t = 2 (2 ± 0.45).Check your understandingQuiz1. For equally spaced x-values, a relation is quadratic when:its second differences are constant and not zeroits y-values double each timeits graph is a straight lineits first differences are constant2. Second differences of −4 tell you the parabola:opens upis a straight linehas no vertexopens down3. A horizontal line at h = 12 crosses a ball's height graph at t = 0.8 s and t = 3.2 s. When is the ball highest?t = 2.4 st = 2 st = 4 st = 1.6 s4. A projectile's height graph has zeros at t = 0 and t = 6. What do the zeros mean?the starting heightthe speedthe times of maximum heightthe times when the height is 0 (launch and landing)5. Which question can the soccer-ball height graph answer?Who kicked it?When was the ball 16 m high?What colour was the ball?How heavy was the ball?6. Values read from a hand-drawn graph are best described as:estimatesalways wrongalways integersexactMark lesson completeFor parents and teachersBig idea (A1.1): students pose and solve problems about quadratic relations given as tables of values and graphs, before any algebra. This lesson also reviews Grade 10 ideas (second differences, vertex, zeros) that the rest of the unit relies on.How to help: ask your teen to explain a graph to you in plain words: What does this point mean? When does the ball reach its highest point? Why are there two answers? Then ask them to make up a question for you to answer from the graph.Common difficulties:Joining points with straight segments instead of a smooth curve.Giving only one of the two times when a horizontal line meets the parabola.Forgetting units, or giving answers outside the real situation (negative time).Computing first differences with uneven x-steps. The x-values must be equally spaced for the differences test to work.NextExpanding and Simplifying Quadratic Expressions