Tasks | Early Adolescence | Equations of Quadratics

Tasks - Year 10

Equations of Quadratics   by Peter Cazalet, Collie SHS

Phase of Development:   Early Adolescence

Learning Area/s:	Mathematics
Strand/s	Substrand/s
Algebra	Understanding Graphs Represent Variation
Working Mathematically	Reason Mathematically
Brief Description:	Using the Stat Menu of the Graphics Calculator to find function rules, then solve problems related to the rule found using the Calculators G-solve capabilities.
Expected Outcomes:	Students should establish minimum conditions to find the function rule for any quadratic function and then apply this understanding to problem solving.
Context:	Year 10, Senior High School, 20 Casio CFX9850 Plus Graphics Calculators.
Learning Activities/ Experience:	Focus Question: How many points do we need to determine the equation of any quadratic function? Assume students know that given two points, we can determine the equation of a line, including using the Graphics Calculator. Using the Stat Menu of the calculator, students check the number of points required in order to find the equation of a quadratic function. Once they establish the number of points is 3 (unless one is the turning point) they can check their claim. Students then conclude that given any three points (not on a straight line) they can use their calculator to find the function rule. Once they have this rule they can answer a number of related "application questions" using the G-Solve capabilities of the calculators.
Resources:	Learning Technologies: Casio CFX 9850G Graphics Calculators.
Practical Frameworks/ Templates/ Worksheets:	Just pose the question and let students find the answer. So, students Enter ordered pairs into Stat Menu Find quadratic regression curve in the form y = ax2 + bx + c Check function rule algebraically Copy from Stat Menu to Graph menu Solve associated problems.
Student Examples:	Examples Note: This is a selection of questions on the Quadratic Function in its various forms. The problems from that list relevant to this task are: 1, 3, 9, 10, 11, 12, 14, 16 and 17.
Assessment & Evaluation:	It is not necessary to formally assess; students working through follow up questions should be able to demonstrate that they have achieved the outcomes.

Examples

The Quadratic Function

1. A shell is fired from an artillery gun. The shell follows a parabolic arc with a range of 800 metres (on flat ground) reaching a maximum height of 50 metres, halfway to the target.
   If the origin is defined as the point of firing, find the quadratic function defining the shell's path,
   (a) give three points the parabolic arc contains.

   (b) using the statistical mode of your calculator, plot the points and hence graph the quadratic function.

   (c) state the function defining the shell's path.

   A tall building is 600 metres from the point of firing.

   (d) If the building is 35 metres high, show that the shell will fail to hit the building.

   For every 100 metres reduction in the shell's range, the artillery gun's adjustment results in an extra 15 m in the shell's maximum height.

   (e) If the gunner's aim is to destroy the building by striking it with a shell at the building's base, what function should determine the gun's shell path?

2. A shell is fired from an artillery gun. The shell follows a parabolic arc with a range of 800 metres (on flat ground) reaching a maximum height of 50 metres, halfway to the target.
   If the origin is defined as the point of firing, find the quadratic function defining the shell's path and express the function defining the shell's path in the form:
   (a) S(x) = A (x = p)(x +q)

   (b) S(x) = a(x + p)²

   (c) S(x) = ax² + bx + c

3. Mr. Hunter hits a golf ball 150 metres on flat ground with a seven-iron. The path of the ball's flight is parabolic reaching a maximum height of 28 metres halfway to the target. Mr. Hunter sees a tree immediately in front of, but 25 metres from the green (his target). He estimates that the tree is 15 metres away.
   (a) Use your calculator to show that the shot will clear the tree.

   (b) Justify your answer algebraically.

4. A sky rocket is fired from the ground and its height h (in metres) at time t (seconds) after launch is given by the equation h(t) = 16t - t²
   (a) How high is the rocket after
   (i) 5 seconds
   (ii) 12 seconds

   (b) When will the rocket reach its maximum height and what is this height?

   (c) When will the rocket hit the ground?

   (d) Graph the function and use your graph to determine when the rocket is 30m above the ground (to 1 decimal place).

   (e) Use an algebraic technique to confirm your answer to part (d).

5. Tyrone Shoelaces, a rock climber, is resting on a ledge on a vertical cliff. He throws a stone to the opposite side of a stream at the base of the cliff. The path of the stone is parabolic with the equation.

   Y = -2(x+2)(x - 8)

   At the peak of the stone's flight it is level with the top of the cliff and it hits the ground just on the other side of the stream. - let the x axis represent the ground from the cliff in metres. - let the y axis represent the cliff height in metres.

   (a) How high is the ledge above the ground?

   (b)How high is the cliff?

   (c) What is the distance from the base of the cliff to the far side of the stream?

6. Constance (Connie to her friends) Struction is an engineer designing a suspension bridge. She decides that the supporting pylons will be 80 m apart and that the supporting cable satisfies the equation y = 0.01x² + 20.

   The road satisfies the equation y = 15 while the river is y = 0.

   (a) What is the height of the road above the river?

   (b) What is the height of the lowest point of the supporting cable above the river?

   (c) What is the height of the pylon?

   (d) Draw a diagram showing the span of the bridge between the two pylons.

   (e) If the vertical supporting cables are 10 m apart, starting from the lowest point on the parabola, what is the total length of this cable required for one span of the bridge?

7. Old McDonald has a farm and wishes to make a sheep pen, rectangular in shape, against an existing fence line. He has 100 m of fencing and wishes to maximise the enclosed area. He decides to lex x be the width of the pen as illustrated below.

   existing fence line
   

   (a) Express the length l in terms of x.

   (b) Find an equation for the area of the enclosure (A) in terms of x.

   (c) What type of function is y = A(x)?

   (d) Graph the function with x on the horizontal axis and A(x) on the vertical axis.

   (e) What value of x gives the maximum area?

   (f) What is the maximum area?

8. Find the equations of the following parabolas:

   A parabola which has-

   (a) a turning point at (5,4) and contains the origin.

   (b) a y-intercept of 3 and a turning point at (2, -3).

   (c) a turning point at (2, 4) and intersects the x axis at x = 5 and x = 1.

   (d) a maximum value of 7 when equal to -1 and containing (1,3).

   NOTE: Problems of this type follow the same procedure
   - Use the turning point to find ____________________
   - Use any point apart from the turning point to find ____________

   We can then determine the equation in the form y = a(x + p)² = q

9. Two 25 m power poles PQ and TV stand on horizontal ground 150 m apart. The cable forms a parabolic arc with its lowest point 17 m above the ground, and lies midway between the poles. Find the equation of the parabola if

   (a) M is the origin (d) T is the origin

   (b) Q is the origin (e) V is the origin

   (c) P is the origin

   

10. A golfer hits a golf ball 176m with a five-iron. The ball follows a parabolic arc reaching a maximum height of 29 m midway to the target (the green).
    (a) Find the equation of the parabolic arc.

    (b) If there is a tree 18 metres high directly between the golfer and the green, what is the minimum distance from the tree that the golfer can hit the shot and still clear the tree?

11. Two buildings are on opposite sides of a 20 metre wide street. A power cable is strung between the buildings as indicated in the diagram below. The lowest point of the cable is 4 metres above the ground and three fifths of the way across the intersection. If the tall building is 32 metres high, how high is the shorter (lower) building?

    

12. A parabolic archway sits on top of two 3 m pillars which are 10 m apart. The top of the arch is 6 m high and midway between the pillars.
    (a) If O is the origin, find the equation of the arch.

    (b) Would a bus 4 m wide and 4 m high pass under this arch? Justify your answer with the aid of a sketch if you use your calculator.

    

13. The water level of a North West tidal harbour on a particular afternoon is given by

    h =

    t (10 - t ) 10

    Where t is the number of hours after 12.00 noon and h is the height of the tide, in metres, after 12.00 noon

    (a) When is high tide?

    (b) How high is high tide?

    (c) If a ship can enter the harbour and only remain in the harbour when the tide is at least 2 metres above the noon level, how long can the ship remain in the harbour?

14. A firefighter holding a fire hose is standing on top of a ladder 5 m above the ground. A fuel tank with a 14 m high wall is ablaze 15 m from the point X directly below the firefighter (see diagram below).

    The arc of the water from the hose follows a parabolic arc reaching a maximum height of 18 m, ten metres from X. (ie. 10 m along the ground).

    (a) Find the equation of the arc of water if X is the origin.

    (b) Given this equation, will the path of the water clear the wall of the fuel tank?

    

15. The diagram below represents the estimated number of strawberries, f (in hundreds), which ripened t hours after noon on a particular farm.

    

    (a) What model (function) do these points represent?

    (b) Find the function rule that will allow f to be calculated given t.

    (c) When is the optimum picking time?

    (d) If picking is delayed until 3.30pm, how many ripe strawberries could we expect to pick?

    (e) If the strawberries were picked at 11.00am, how many ripe strawberries would the farmer expect?

    (f) At what time would the total crop be ruined?

16. (a) How many points are required to use the calculator to find the equations of a quadratic function?

    (b) When are two points sufficient to find the equation of a quadratic function?

17. When Tom Moody throws a ball from the outfield he releases it from a height of 2 metres. The ball's flight path is part of a parabolic arc hitting the ground 30 metres from the point Tom threw the ball.
    (a) Given the above information, we can not find the equation of the parabolic arc. Why?

    (b) What condition would have to apply to enable us to find the equation of the parabola given the above information?

    The ball just clipped the bail (without deviating in flight) on top of an 80 cm wicket which was 3 metres from the point the ball hit the ground.

    (a) Can you find the equation of the parabolic arc? Explain.

    (b) What is the maximum height of the ball?

    (c) How far from the wicket was the ball when it reached its maximum height.

    (d) If Justin Langer who is 1.77 cm tall was standing on a line between where Tom threw the ball and the wicket, what is the range of distances from Tom that Justin could be standing and not be struck by the ball?

     

    [ TOP ] [ HOME ] [ SITE MAP ]

    ---

    Copyright © Department of Education, Western Australia. All Rights Reserved.
    updated January 2002