C ontent and L anguage I ntegrated L earning

1. Content andLanguage Integrated Learning

2. Quadratic functions OBJECTIVES: • to arouse interest and curiosity showing images and introducing real problems involving quadratic functions. • - to know quadratic functions

3. Images and real problems involving quadratic functions Galileo proved that the trajectory of a projectile travelling through a non resisting medium is a parabola (1608) A projectile is an object upon which the only force is gravity. A vertical force does not effect a horizontal motion. The result of a vertical force acting upon a horizontally moving object is to cause the object to deviate from its otherwise linear path

4. Parabolic shapes

5. Quadratic functions

6. Sketching a quadratic graph with Geogebra

7. Parabola as a locus • Activity: construction of a parabola as a locus with Geogebra • Instructions to sketch the graph of a parabola as a locus with Geogebra: • Plot a straight line d, called “directrix” • Plot a point F which doesn’t lie on d, called “focus” • Plot a point Q on d • Plot the axis a of the line segment FQ • Plot the straight line b through Q and perpendicular to d • Plot the point P, intersection of a and b • Plot the locus of P with respect to Q on d • The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus.

8. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus.

9. Quadratic inequalities

10. Real life problems

11. Real Life problems A farmer has 16 metres of fencing with which to make a rectangular enclosure for sheep. If one side of the enclosure is xmetres long, show that the area A is given by Draw the graph of A(x) in the domain . Use your graph to estimate: • The area of the enclosure when x = 2.5 • The maximum possible area and the value of x when this occurs • The two values of x for which the area is 12 m2. Jenny fires a scale model of a TV talent show presenter horizzontally with a velocity of 100 ms-1 from 1.5m above the ground. • How long does it take to hit the ground, and how far does it travel? • Assume the model acts as a particle, the ground is horizontal and there is no air resistance.

12. Real-lifeProblem • Draw the graph of the function y=6x- x² in the domain 0 ≤ x ≤ 6. • Y is the height, in metres, reached by a golf ball from the time it was hit (x=0) to the time it hits the ground (x=6). If each unit on the x-axis represents 1 second and each unit on the y-axis represents 5 metres, use your graph to estimate: • the greatest height reached by the golf ball • the height of the golf ball after 1.5 seconds

13. Howto sketch the graph Y=0 Y=x²-6x C.E. 0 ≤ x ≤ 6 Y= 6x-x² -6+x²=0 x(x-6)=0 x₁=0 x₂=6 Xv=-b/2a= 6/2= 3 Yv=6(3) – (3)²= 9

14. Parabola’sgraph

15. Process 1.Estimate the greatest height reached by the golf ball: Y= 9 9x5m= 45 m 2. Estimate the hight of the golf ball after 1.5 seconds: Y= 6(1.5) – (1.5)²= 6.75 Hight: 6.75x5m= 33.75m

16. Classwork

17. Self assessment