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Exploring Equations: Graphical Solutions and Functions Shape

Learn how to solve equations using graphs, analyze quadratic, cubic, and reciprocal functions, and find values and intervals. Improve your graphing skills to understand mathematical relationships visually.

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Exploring Equations: Graphical Solutions and Functions Shape

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  1. Unit 29Using Graphs to Solve Equations

  2. Unit 29 29.1 Solution of Simultaneous Equations Using Their Graphs

  3. Example Solve the pair of simultaneous equations by drawing their graphs Solution We can rewrite each equation in the form y = ..... Plotting the lines, when when Plot these points Intersection at y ? ? ? ? ? ? ? x ? ? ?

  4. Unit 29 29.2 Graphs of Quadratic Functions

  5. Quadratic functions contain an x² term as well as multiples of x and a constant. The following graphs show 3 examples. y y x y y 0 0 0 x x x

  6. Discuss the shape of the examples below. y y y 0 0 x 0 x x

  7. Unit 29 29.3 Graphs of Cubic Function

  8. Cubic functions involve an x³ term and possibly x², x and constant terms as well. The graphs below show some examples • The graph of a cubic function can: • Cross the x-axis once as in example (a) • Touch the axis once as in example (b) • Cross the x-axis three times as in example (c) y y y 0 0 0 x x x

  9. Unit 29 29.4 Reciprocal Functions

  10. Reciprocal functions have the form of a fraction with x as the denominator. The graphs below show some examples The curves are split into two distinct parts. The curves get closer and closer to the axis, as . The curves have two lines of symmetry, and . y y y 0 0 0 x x x

  11. The curves are split into two distinct parts. The curves get closer and closer to the axis, as . The curves have two lines of symmetry, and . Where on the grids below would the stated curves lie. y y 0 0 x x

  12. Unit 29 29.5 Graphical Solutions of Equations

  13. Use the graph to determine: (e) The value of at which is a minimum (f) The interval on the domain for which is less than • (a) The value of when • (b) The value of when ? (c) The value of when or (d) The minimum value of ? ? ? ? ? ? ? y 0 x

  14. (a) Given that , complete this table. • (b) Graph this equation for ? ? ? ? (c) Use this graph to solve Draw the line on the grid. This intersects at or

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