Thèmes / GCSE / Mathematics / Algebra Quadratic graphs and their properties Apprends en jouant
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Notes de leçon The shape of a quadratic graph A quadratic has the form y = ax2 + bx + c with a \neq 0. Its graph is a smooth, symmetrical curve called a parabola . If a > 0 the parabola is U-shaped with a minimum; if a < 0 it is ∩-shaped with a maximum. The y-intercept is c, where x = 0. Positive and negative leading coefficients Positive and negative leading coefficients O −6 −4 −2 2 4 6 8 10 −3 −2 −1 1 2 3 x y y = 3 − x² y = x² Roots and the line of symmetry Real roots are the x-values where the graph meets the x-axis, so y = 0. It may cross the axis or touch it at a repeated root. The parabola is symmetrical about a vertical line of symmetry halfway between the roots: x = \frac{x1 + x2 }{2}, or x = -\frac{b}{2a}. A quadratic may have two roots, one repeated root (the graph just touches the axis), or no real roots (the graph misses the axis). Roots and turning point Roots and turning point O −2 2 4 6 8 10 −1 1 2 3 4 5 x y (1, 0) (3, 0) y = x² − 4x + 3 Min (2, −1) The turning point The turning point (vertex) lies on the line of symmetry. Substitute that x-value into the equation to find y. For y = x2 - 4x + 3: roots at x = 1 and x = 3, line of symmetry x = 2, turning point y = 4 - 8 + 3 = -1, so (2, -1). Completed square form: y = (x - p)2 + q has turning point (p, q). Drawing a quadratic Make a table of values for several x values (include negatives), plot the points and join them with a smooth curve, not straight lines. Check the curve is symmetrical and passes through the intercepts. Diapos Sign up free to view the lesson slides
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Questions d'entraînement Aperçu gratuit — 8 sur 34 questions. Inscris-toi pour toutes les voir. 1. What is the graph of a quadratic function called?
Easy A A hyperbolaB A parabolaC A circleD A straight line2. For y = x2 - 5, where does the graph cross the y-axis?
Easy A (0, 5)B (0, -5)C (5, 0)D (-5, 0)3. What shape is y = 2x2 + 1?
Easy A ∩-shapedB U-shapedC a straight lineD an S-shape4. What shape is y = 3 - x2 ?
Easy A U-shapedB ∩-shapedC a straight lineD a circle5. What are the roots of y = x2 - 9?
Easy A x = 3 onlyB x = \pm 3C x = \pm 9D x = 06. The roots of a quadratic are x = 1 and x = 5. What is the line of symmetry?
Easy 7. The roots of a quadratic are x = -2 and x = 6. What is the line of symmetry?
Easy 8. What is the line of symmetry of y = x2 + 6x + 1?
Easy A x = 3B x = -3C x = 6D x = -6Plus de questions 9. What is y when x = 3 on y = x2 - 2x?
Easy 10. What is the turning point of y = x2 + 2?
Easy A (2, 0)B (0, 2)C (0, -2)D (1, 3)11. What is the turning point of y = x2 - 4x + 3?
Medium A (2, -1)B (2, 1)C (-2, -1)D (4, 3)12. What is the turning point of y = (x + 1)2 - 4?
Medium A (1, -4)B (-1, -4)C (-1, 4)D (1, 4)13. The turning point of y = x2 - 6x + 5 is:
Medium A (3, -4)B (3, 4)C (-3, -4)D (6, 5)14. How many real roots does y = x2 + 1 have?
Medium 15. How many roots does a parabola that just touches the x-axis have?
Medium 16. A parabola has its minimum at (2, -3). How many roots does it have?
Medium 17. Which equation could match a graph with roots at x = 2 and x = 5 that is U-shaped?
Medium A y = (x - 2)(x - 5)B y = (x + 2)(x + 5)C y = -(x - 2)(x - 5)D y = x2 + 1018. Solve x2 - 5x + 6 = 0 to find where y = x2 - 5x + 6 crosses the x-axis.
Medium A x = 2, 3B x = -2, -3C x = 1, 6D x = 5, 619. What is the minimum value of y = x2 - 6x + 11?
Medium 20. What is the maximum value of y = 5 - (x - 1)2 ?
Medium 21. The graph of y = ax2 + bx + c crosses the y-axis at (0, 7). What is c?
Medium 22. When plotting a quadratic from a table of values, you should join the points with:
Easy A ruled straight linesB a smooth curveC a zigzagD dotted lines only23. Which x value is NOT needed in a table for y = x2 - 2x, from -2 to 4?
Easy 24. Which of these quadratics has y always positive?
Medium A y = x2 - 4B y = x2 + 4C y = x2 - 4xD y = -x2 + 425. y = x2 is symmetrical about which line?
Easy A y = xB the y-axisC the x-axisD x = 126. A quadratic graph is a straight line.
Easy True or false?
27. If the coefficient of x2 is negative the graph has a maximum point.
Easy True or false?
28. The real roots of a quadratic are the x-values where its graph meets the x-axis, either crossing or touching it.
Easy True or false?
29. A quadratic always has two real roots.
Easy True or false?
30. The turning point lies on the line of symmetry.
Easy True or false?
31. y = (x - 2)2 + 3 has its turning point at (-2, 3).
Medium True or false?
32. Which statements about y = x2 - 4x + 3 are true?
Medium A It crosses the x-axis at 1B It crosses the x-axis at 3C Its turning point is a maximumD It crosses the y-axis at (0, 3)E Its line of symmetry is x = 333. Which graphs are ∩-shaped?
Medium A y = -x2 B y = x2 - 1C y = 4 - x2 D y = 2x2 E y = -3x2 + x34. Find the turning point of y = x2 - 6x + 5 and say whether it is a maximum or minimum.
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