Graphical Methods For Solving Equations
Bayani Gaba-gaba
Some equations are too complex to solve with algebra alone. A cubic tangled with a trigonometric function, or an exponential set equal to a polynomial, may have no neat closed-form answer. Graphical methods let you find solutions visually, locating the points where two curves cross and reading off coordinates with whatever precision the question demands.
In this topic you will learn to rewrite a target equation so that its solutions correspond to intersection points on a graph you have already drawn. You will practise choosing the right companion curve, building a table of values, and estimating roots to a given degree of accuracy. These skills appear frequently in the Edexcel 4PM1 papers, where a reference curve is provided and you must add a second line to find the solutions.
Manufura
- Solve equations and transcendental functions by graphical methods
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- The graph of y = x^3 is drawn. To solve x^3 = 2x + 1 graphically, which line should be drawn? A) y = 2x + 1 B) y = x^3 - 2x - 1 C) y = x + 1 D) y = 2x - 1 Answer: A
- Two curves intersect at exactly two points. How many real solutions does the corresponding equation have? A) 0 B) 1 C) 2 D) Infinitely many Answer: C
- Which of the following is a transcendental equation? A) x^3 - 4x + 1 = 0 B) 2^x = 3x + 1 C) x^2 - 5x + 6 = 0 D) x^4 - 1 = 0 Answer: B
- The graph of y = sin x is drawn. To solve sin x = 0.5x, what companion line is needed? A) y = 0.5 B) y = 0.5x C) y = x D) y = 2x Answer: B
- If f(1.4) = -0.2 and f(1.5) = 0.3 for a continuous function f, what can you conclude? A) There is a root between x = 1.4 and x = 1.5 B) The function has no roots C) The root is exactly x = 1.45 D) The function is not continuous Answer: A
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