AQA · Mathematics · 8300
GCSE Maths: Change of Sign
Algebra · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths practice questions for Change of Sign (Corbettmaths Video 373), and every card links back to the exact page it came from. Practice tests are drawn from this deck’s own cards. Source material © Corbettmaths — used with attribution.
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What does a change of sign in f(x) between x = a and x = b tell you?
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A root of f(x) = 0 lies between a and b (as long as f is continuous there).
Between which two consecutive integers does x³ − 8x − 10 = 0 have a solution?
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x = 3 and x = 4. f(3) = −7 and f(4) = 22, so there is a change of sign.
How do you show x³ + 2x = 1 has a solution between x = 0 and x = 1?
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Let f(x) = x³ + 2x − 1. f(0) = −1 and f(1) = 2; the change of sign means a root lies between 0 and 1.
What do the values x₁, x₂, x₃ from an iteration formula represent?
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Successive approximations (estimates) to a root of the original equation, each usually closer to the solution than the last.
How do you obtain an iteration formula from an equation f(x) = 0?
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Rearrange it into the form x = g(x), then write xₙ₊₁ = g(xₙ).
In an iteration xₙ₊₁ = g(xₙ), each new estimate is found by substituting the ? estimate into the formula.
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Why does the choice of starting value x₀ matter in an iteration formula?
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Different rearrangements converge to different roots or diverge; x₀ should be close to the root you want so the iteration converges to it.
What does it mean for an iteration to converge?
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The successive values xₙ get closer and closer to a fixed value, which is a root of the equation.
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