Graphs and Functions

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    Kautsar Cakrawala MargoLuqman AlfarisiNursabrinah MutiarasariZephania Novia M.

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    Is a delicate combination of algebra

    and geometery.

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    In line or curve plotting of a graph, we normally come across the term"Gradient". This is an important parameter that governs the profile of theplotted line or curve.

    So, what is this "gradient"?

    Gradient is defined as the ratio of the change in the vertical unit tothe change in the horizontal unit (at a certain point on the curve).

    From the definition, it implies that the larger the numerical value of thegradient, the steeper is the curve or line plotted.

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    Y

    B

    A

    X

    0

    -Y axis representvertical line

    -X axis represent thehorizontal line

    -Line A to B is thegradient, it is a constantline.

    -The line ABjusttouches the curve- itdoes not cross it. So thisgradient is called atangent.

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    POSITIVE GRADIENT

    Positive gradient exhibitsvertically upward

    increment for increase inhorizontal directiontowards positive "x" value.

    NEGATIVE GRADIENT

    Negative gradient shows adownward tendency whenmoving towards the positive "x"direction.

    For straight line equation of y =mx + c , "m" represents thegradient.

    Example 1: y = 3 x + c means apositive gradient with upward-going line.

    Example 2: y = -4 x + c means a

    negative gradient withdownward-going line.

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    Positive Gradient

    Diagram of a PositiveGradient

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    Negative Gradient

    Diagram of a NegativeGradient

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    However, do note also that forstraight line,

    the gradient is constantas the gradient doesnot change along the line.

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    To find the gradient of a curve, you mustdraw an accurate sketch of the curve. At

    the point where you need to know thegradient, draw a tangentto the curve. Atangent is a straight line which touches

    the curve at one point only. You then findthe gradient of this tangent.

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    It is also possible to find the gradient of a curve at agiven point - thought the gradient (steepness) willchange as you move along the curve, so your answer

    will only be true for the gradient at the point you use. To find the gradient of a curve, you must draw an

    accurate sketch of the curve. At the point where youneed to know the gradient, draw a tangentto thecurve. A tangent is a straight line which touches thecurve at one point only. You then find the gradient ofthis tangent like you did in the section above.

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    Find the gradient ofthe curve y = x at thepoint (3, 9).

    Gradient of tangent = (change in y)/(change in x)

    = (9 - 5)/ (3 - 2.3)= 5.71

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