different types of Graphs for Year10.pptx
- 1. GRAPHS
- 2. • Direct proportion is a type of proportionality relationship. For direct proportion, as one value increases, so does the other value and conversely, as one value decreases, so does the other value. • The symbol ∝ represents a proportional relationship. • If y is directly proportional to x, we can write this relationship as y x ∝ • Direct proportion is useful in numerous real-life situations, such as exchange rates, unit conversion, and fuel prices. Direct Proportion
- 3. If y is directly proportional to x, then y = k, where k is a constant (number) called the constant of proportionality or constant of variation. A direct linear relationship exists between and y. If increases (or decreases), y increases (or decreases) If is doubled (or halved), y is doubled (or halved) Another way of saying 'y is directly proportional to ' is y varies directly with ‘ The graph of direct proportion is a straight line going through (0, 0) with gradient k Summary
- 4. The cost of a circular table is directly proportional to the square of the radius. " A circular table with a radius of 40cm cost $50. " What is the cost of a circular table with a radius of 60cm? " Problems
- 5. Problems
- 6. Ex 7:01
- 7. • Two variables are inversely proportional to each other if, when one variable increases, the other one decreases by the same factor. • An example of inverse proportion would be the hours of work required to build a wall. If there are more people building the same wall, the time taken to build the wall reduces. Inverse Proportion
- 8. Inverse Proportion If y is inversely proportional to x, then where k is a constant (number) called the constant of proportionality or constant of variation. •If x increases, y decreases ('inverse' means 'opposite’) •If x decreases, y increases •If x is doubled, y is halved •If x is halved, y is doubled •Another way of saying 'y is inversely proportional to x' is 'y varies inversely with
- 9. The time (t) in minutes taken by a car to travel 120 km is inversely proportional to the speed (s in km/h) of the car. At 60 km/h, the time taken is 120 minutes. a. Find the inverse variation equation for t. b. How long did the car take to travel 120 km at: i. 40 km/h? ii. 110 km/h? c. Find the car's speed if it took 45 minutes to travel 120 km. Problems
- 10. The temperature, T (in degrees Celsius), of the air is inversely proportional to the height, h (in meters), above sea level. At 800 m above sea level, the temperature is 10°C. Questions: a) What is the temperature at 1500 m above sea level? b) Graph the relationship between temperature and height above sea level for heights between 0 and 5000 m. Problems
- 11. Two people take 15 minutes to clean the room. How many minutes will three people take to clean the same room? Problems
- 12. Exercise 7.02
- 13. Conversion graphs Conversion graphs are straight line graphs that show a relationship between two units and can be used to convert from one to another. They are very useful to solve real-life problems. Some conversion graphs can show a direct proportion between two units, for example, converting between two currencies to show an exchange rate, such as pounds sterling (British pounds) to US dollars.
- 14. 1.Locate the values on the axis representing the unit you wish to convert. 2.Draw a perpendicular line from the value on the axis to the conversion line. 3.Draw a line from the conversion line perpendicular to the other axis and read off the conversion value. How to use conversion graphs
- 18. Exercise 7.03
- 19. Parabola
- 20. Quadratic Function • A function of the form y=ax2 +bx+c where a≠0 making a u-shaped graph called a parabola. - If a is positive, u opens up - If a is negative, u opens down • The highest exponent of x is 2
- 22. 𝑻𝒉𝒆𝒈𝒓𝒂𝒑𝒉𝒐𝒇 𝒚=𝒂𝒙𝟐 For the graph of a quadratic equation in the form , where “a” is a constant (number), the size of a (the coefficient of ) affects whether the parabola is 'wide' or 'narrow’. As the size of “a” increases, the parabola becomes 'narrower' and as the size of a decreases, the parabola 'widens'. If a is negative, then the parabola is concave down.
- 23. + C •For the graph of a quadratic equation in the form y = + c, where a and c are constants, the effect of c is to move the parabola up or down from the origin. Also, c is the y-intercept of the parabola.
- 24. Graph each set of quadratic Equations, showing the vertex of each parabola +5 ,
- 25. Graph each set of quadratic Equations, showing the vertex of each parabola ,
- 27. Graphs of Exponential Functions
- 29. Note that in the definition of an exponential function, the base a = 1 is excluded because it yields f (x) = 1x = 1. This is a constant function, not an exponential function. Exponential Functions
- 30. Real-life examples of exponential graph
- 31. Graphs of y = ax In the same coordinate plane, sketch the graph of each function by hand. a. f(x) = 2x b. g(x) = 4x Solution: The table below lists some values for each function. By plotting these points and connecting them with smooth curves, you obtain the graphs shown in Figure 3.1. Figure 3.1
- 32. Note that both graphs are increasing. Moreover, the graph of g(x) = 4x is increasing more rapidly than the graph of f(x) = 2x . You can tell if you compare the y values in the table below. cont’d Graphs of y = ax
- 33. Graph of y = 2x and
- 34. Definition of a Circle A circle is a set of points a given distance from one point called the center. The distance from the center is called the radius Definition of a Circle
- 35. Equation of a circle
- 36. Graph
- 37. Find the equation of a circle with centre(0,0) and diameter 25