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- 1. Quadratic Functions IB Studies, Syllabus ref 4.3 Lesson 2
- 2. Vertex or Turning point
- 3. Vertex or Turning point The vertex of a parabola is where the parabola “turns around”.
- 4. Vertex or Turning point The vertex of a parabola is where the parabola “turns around”. In Lesson 1 we learnt that this occurs on the axis of symmetry.
- 5. Vertex or Turning point The vertex of a parabola is where the parabola “turns around”. In Lesson 1 we learnt that this occurs on the axis of symmetry. Hence the x-coordinate of the vertex will be the same as the AoS and the y- coordinate can be found by substitution or in function notation is:
- 6. Vertex or Turning point The vertex of a parabola is If the value of ‘a’ is positive where the parabola “turns then the parabola is concave around”. up and the vertex would be a minimum turning point. In Lesson 1 we learnt that this occurs on the axis of symmetry. Hence the x-coordinate of the vertex will be the same as the AoS and the y- coordinate can be found by substitution or in function notation is:
- 7. Vertex or Turning point The vertex of a parabola is If the value of ‘a’ is positive where the parabola “turns then the parabola is concave around”. up and the vertex would be a minimum turning point. In Lesson 1 we learnt that this occurs on the axis of symmetry. If the value of ‘a’ is negative then the parabola is concave down and the vertex would Hence the x-coordinate of be a maximum turning the vertex will be the same point. as the AoS and the y- coordinate can be found by substitution or in function notation is:
- 8. Vertex or Turning point The vertex of a parabola is If the value of ‘a’ is positive where the parabola “turns then the parabola is concave around”. up and the vertex would be a minimum turning point. In Lesson 1 we learnt that this occurs on the axis of symmetry. −b If the value of ‘a’ is negative x= then the parabola is concave 2a down and the vertex would Hence the x-coordinate of be a maximum turning the vertex will be the same point. as the AoS and the y- coordinate can be found by substitution or in function notation is:
- 9. Vertex or Turning point The vertex of a parabola is If the value of ‘a’ is positive where the parabola “turns then the parabola is concave around”. up and the vertex would be a minimum turning point. In Lesson 1 we learnt that this occurs on the axis of symmetry. −b If the value of ‘a’ is negative x= then the parabola is concave 2a down and the vertex would Hence the x-coordinate of be a maximum turning the vertex will be the same point. as the AoS and the y- coordinate can be found by substitution or in function notation is: ⎛ −b ⎞ y = f ⎜ ⎟ where f ( x ) = ax 2 + bx + c ⎝ 2a ⎠
- 10. Vertex or Turning point The vertex of a parabola is If the value of ‘a’ is positive where the parabola “turns then the parabola is concave around”. up and the vertex would be a minimum turning point. In Lesson 1 we learnt that this occurs on the axis of symmetry. −b If the value of ‘a’ is negative x= then the parabola is concave 2a down and the vertex would Hence the x-coordinate of be a maximum turning the vertex will be the same point. as the AoS and the y- coordinate can be found by substitution or in function notation is: ⎛ −b ⎞ y = f ⎜ ⎟ where f ( x ) = ax 2 + bx + c ⎝ 2a ⎠
- 11. Vertex or Turning point The vertex of a parabola is If the value of ‘a’ is positive where the parabola “turns then the parabola is concave around”. up and the vertex would be a minimum turning point. In Lesson 1 we learnt that this occurs on the axis of symmetry. −b If the value of ‘a’ is negative x= then the parabola is concave 2a down and the vertex would Hence the x-coordinate of be a maximum turning the vertex will be the same point. as the AoS and the y- coordinate can be found by substitution or in function notation is: ⎛ −b ⎞ y = f ⎜ ⎟ where f ( x ) = ax 2 + bx + c ⎝ 2a ⎠
- 12. Example 1 - Determine the coordinates of thevertex of y = 2x − 8x + 1 2
- 13. Example 1 - Determine the coordinates of thevertex of y = 2x − 8x + 1 2Firstly, match the equation to the general formand write down the values of a, b, and c.y = ax 2 +bx + cy = 2x 2 − 8x + 1hence a = 2 , b = −8 , c = 1 −bnow axis of symmetry equation from x = 2a
- 14. Example 1 - Determine the coordinates of thevertex of y = 2x − 8x + 1 2Firstly, match the equation to the general formand write down the values of a, b, and c.y = ax 2 +bx + cy = 2x 2 − 8x + 1hence a = 2 , b = −8 , c = 1 −bnow axis of symmetry equation from x = 2a − ( −8 ) 8so x = = =2 2×2 4and remember this matches the x-coord of the vertexso now all we need is the y-coord
- 15. Example 1 - Determine the coordinates of thevertex of y = 2x − 8x + 1 2Firstly, match the equation to the general formand write down the values of a, b, and c.y = ax 2 +bx + cy = 2x 2 − 8x + 1hence a = 2 , b = −8 , c = 1 −bnow axis of symmetry equation from x = 2a − ( −8 ) 8so x = = =2 2×2 4and remember this matches the x-coord of the vertexso now all we need is the y-coordby substituting x = 2, in to the parabola 2 y = 2 (2) − 8 (2) + 1 = 8 − 16 + 1 = −7
- 16. Example 1 - Determine the coordinates of thevertex of y = 2x − 8x + 1 2Firstly, match the equation to the general formand write down the values of a, b, and c.y = ax 2 +bx + c Hence the vertexy = 2x 2 − 8x + 1 has coordinateshence a = 2 , b = −8 , c = 1 −b (2,-7)now axis of symmetry equation from x = 2a − ( −8 ) 8so x = = =2 2×2 4and remember this matches the x-coord of the vertexso now all we need is the y-coordby substituting x = 2, in to the parabola 2 y = 2 (2) − 8 (2) + 1 = 8 − 16 + 1 = −7
- 17. Example 1 - Determine the coordinates of thevertex of y = 2x − 8x + 1 2Firstly, match the equation to the general formand write down the values of a, b, and c.y = ax 2 +bx + c Hence the vertexy = 2x 2 − 8x + 1 has coordinateshence a = 2 , b = −8 , c = 1 −b (2,-7)now axis of symmetry equation from x = 2a − ( −8 ) 8so x = = =2 2×2 4and remember this matches the x-coord of the vertexso now all we need is the y-coordby substituting x = 2, in to the parabola Now would this be 2 y = 2 (2) − 8 (2) + 1 a maximum or a = 8 − 16 + 1 minimum turning = −7 point???
- 18. Example 2 - For the quadratic equation y = −x 2 + 2x + 3 . Find:(a) its axes intercepts(b) the equation of the axis of symmetry(c) the coordinates of the vertex(d) sketch the function, labeling the maximum or minimum turning point
- 19. Example 2 - For the quadratic equation y = −x 2 + 2x + 3 . Find:(a) its axes intercepts(b) the equation of the axis of symmetry(c) the coordinates of the vertex(d) sketch the function, labeling the maximum or minimum turning pointa) Firstly, to find the x-intercepts we must factorise!But watch out for the negative, take it out first.y = −x 2 + 2x + 3= − ⎡ x 2 − 2x − 3⎤ ⎣ ⎦= − ⎡( x − 3) ( x + 1) ⎤ ⎣ ⎦hence the x-intercepts are 3 and -1.The y-intercept we can just read from the eqn as 3.
- 20. Example 2 - For the quadratic equation y = −x 2 + 2x + 3 . Find:(a) its axes intercepts(b) the equation of the axis of symmetry(c) the coordinates of the vertex(d) sketch the function, labeling the maximum or minimum turning pointa) Firstly, to find the x-intercepts we must factorise!But watch out for the negative, take it out first.y = −x 2 + 2x + 3= − ⎡ x 2 − 2x − 3⎤ ⎣ ⎦= − ⎡( x − 3) ( x + 1) ⎤ ⎣ ⎦hence the x-intercepts are 3 and -1.The y-intercept we can just read from the eqn as 3.b) The axis of symmetry comes from the coefficientsnow a = −1,b = 2, c = 3 −b − ( 2 )x= = =1 2a 2 ( −1)hence the axis of symmetry is x = 1.
- 21. Example 2 - For the quadratic equation y = −x 2 + 2x + 3 . Find:(a) its axes intercepts(b) the equation of the axis of symmetry(c) the coordinates of the vertex(d) sketch the function, labeling the maximum or minimum turning pointa) Firstly, to find the x-intercepts we must factorise!But watch out for the negative, take it out first.y = −x 2 + 2x + 3= − ⎡ x 2 − 2x − 3⎤ ⎣ ⎦= − ⎡( x − 3) ( x + 1) ⎤ ⎣ ⎦hence the x-intercepts are 3 and -1.The y-intercept we can just read from the eqn as 3.b) The axis of symmetry comes from the coefficientsnow a = −1,b = 2, c = 3 −b − ( 2 )x= = =1 2a 2 ( −1)hence the axis of symmetry is x = 1.c) The vertex comes from substituting x = 1, 2y = − (1) + 2 (1) + 3 = 4hence the coordinates of the vertex is (1,4).
- 22. Example 2 - For the quadratic equation y = −x 2 + 2x + 3 . Find:(a) its axes intercepts(b) the equation of the axis of symmetry(c) the coordinates of the vertex(d) sketch the function, labeling the maximum or minimum turning pointa) Firstly, to find the x-intercepts we must factorise!But watch out for the negative, take it out first.y = −x 2 + 2x + 3= − ⎡ x 2 − 2x − 3⎤ ⎣ ⎦= − ⎡( x − 3) ( x + 1) ⎤ ⎣ ⎦hence the x-intercepts are 3 and -1.The y-intercept we can just read from the eqn as 3.b) The axis of symmetry comes from the coefficientsnow a = −1,b = 2, c = 3 −b − ( 2 )x= = =1 2a 2 ( −1)hence the axis of symmetry is x = 1.c) The vertex comes from substituting x = 1, 2y = − (1) + 2 (1) + 3 = 4hence the coordinates of the vertex is (1,4).
- 23. Now complete these in your books: You only have to do the ﬁrst column here! Click here when you’ve completed them and marked them all!
- 24. Last ones for today...You only haveto do the ﬁrst column here! Click here when you’ve completed them and marked them all!
- 25. Well done! I’m impressed!

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