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C1: Chapters 1-4 Revision

C1: Chapters 1-4 Revision. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 10 th October 2013. Solving simultaneous equations. Remember that the strategy is to substitute the linear equation into the quadratic one, then solve. ?. Expanding out correctly!. ?. Inequalities.

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C1: Chapters 1-4 Revision

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  1. C1: Chapters 1-4 Revision Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 10th October 2013

  2. Solving simultaneous equations Remember that the strategy is to substitute the linear equation into the quadratic one, then solve. ?

  3. Expanding out correctly! ?

  4. Inequalities • For inequalities in general: • Multiplying/dividing both sides by a negative number flips the inequality. • Don’t mix up AND and OR. “” is different from “”. Remember for quadratic inequalities: Always start by putting in the form or .If you have , ABSOLUTELY DON’T divide by , but write Then factorise. Then sketch.Your answer will either be , or “or”. Be sure to use the word ‘or’ in the latter one, since ‘and’ would be wrong. Find the set of values of x for which (a) 4x – 3 > 7 – x (b) 2x2 – 5x – 12 < 0 (c) both 4x – 3 > 7 – x and 2x2 – 5x – 12 < 0 ? ? ?

  5. Discriminant • Whenever you see the words “equal roots”, “distinct/different roots” or “no roots”, you know you’ve got to calculate the discriminant, which is . • It helps to explicitly write out your , and first before substituting into the discriminant. • Be VERY careful with double (or even triple!) negatives.The discriminant of is .The discriminant of is 4. • When you have ‘different roots’ or ‘no roots’, you’ll have a quadratic inequality. Solve in the same way as before. But remember your sketch is in terms of , not in terms of the original variable . So don’t be upset if your sketch has roots, even if the original question asks where your equation has no roots. ? ? The equation , where k is a constant, has 2 different real solutions for x. (a) Show that k satisfies (b) Hence find the set of possible values of k. ?

  6. Sketching quadratics/cubics • For cubics, think whether the / term is positive or negative. Cubics with positive will go uphill, and downhill otherwise. • If , without fully expanding you can tell you’ll have a term, thus it goes downhill. Be careful though: in , the term will be positive! • You can get the roots/-intercepts by setting to be 0. Imagine each factor/brackets being 0. So if , then the roots are • For both quadratics and cubics, the curve touches the x-axis for a root if the factor is squared, and crosses if not repeated. • Don’t forget the y-intercept!YOU WILL LOSE MARK(S) OTHERWISE. • It’s quite acceptable to have algebraic expressions as roots/y-intercepts. The y-intercept is ? No problem! • Don’t forget what a sketch of or looks like.

  7. Sketching cubics Sketch the following, ensuring you indicate the values where the line intercepts the axes. y = (3-x)3 1 y = (x+2)(x-1)(x-3) 5 y = x(x+1)2 9 ? 27 ? ? 6 3 -1 -2 1 3 2 y = x(x-1)(2-x) 6 y = x(1 – x)2 10 y = (x+2)2(x-1) ? ? ? 1 -2 1 2 1 -4 7 y = -x3 11 y = (2-x)(x+3)2 3 y = x(2x – 1)(x + 3) ? ? ? 18 -3 2 0.5 3 4 y = x2(x + 1) 8 y = (x+2)3 12 y = (1 – x)2(3 – x) ? ? ? 8 3 -2 -1 1 3

  8. Transforming Existing Graphs a f(bx + c) + d Bro Tip: To get the order of transformations correct inside the f(..), think what you’d need to do to get from (bx + c) back to x. Step 1: ?  c Step 3: ? ↕ a Step 4: ↑ d ? Step 2: ? ↔ b

  9. Transforming Existing Graphs Here is the graph y = f(x). Draw the following graphs, ensuring you indicate where the graph crosses the coordinate axis, minimum/maximum points, and the equations of any asymptotes. y Bro Tip: Don’t get to transform the asymptotes! This horizontal asymptote won’t be affected by any transformations, but will by ones. y = f(x) (2, 3) 1 x y = 2f(x+2) y = -1 ? y y y y = 0 y = -f(-x) – 1 x 6 1 ? y = f(2x) -2 ? x x (1, 3) y = -1 y = -2 (-2, -4)

  10. Sketching Graphs by Considering the Transform It’s often helpful to consider a simpler graph first, e.g. or , and then consider what transform we’ve done. Sketch ? Start with Then clearly we’ve replaced with and added 4 to the result. i.e. ?

  11. Sketching Graphs by Considering the Transform Sketch ? y x -2 -0.5

  12. Sketching Graphs by Considering the Transform Sketch ? y x

  13. Sketching Quadratics Sketch y = x2 + 2x + 1 Sketch y = x2 + x – 2 y ? y ? 1 x x -2 1 1 -2 Sketch y = -x2 + 2x + 3 Sketch y = 2x2 – 5x – 3 y y ? ? 3 x x -1 3 -0.5 3 -3

  14. Sketching Quadratics Some quadratics have no roots. In which case, you’ll have to complete the square in order to sketch them. This tells you the minimum/maximum point. e.g. So minimum point is . -intercept is 3 ? ? ? Sketch y = -x2 + 2x – 3 Sketch y = x2 – 4x + 5 y ? y ? 5 (1,-2) (2, 1) -3 x

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