Is the Gradient the Same as the Slope in Linear Functions?

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In summary, the gradient of a function is defined as a vector field that assigns a vector to each point of the function. The components of the vector indicate how much the function is changing in that direction. For a linear function like y-3x=2, the gradient is constant and given by (-3,1). However, if the function is more complicated, the components of the gradient may vary. The gradient vector is not necessarily perpendicular to the slope of the function, as for linear functions they are equivalent.
  • #1
SleSSi
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how do u find the gradient of y-3x=2 :confused:
 
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  • #2
in terms of calculus, the gradient is defined to be a vector field, that is, given a function it will assign a vector to each point of the function. the components of each vector tell how much the function is changing in that direction.

[tex]grad(f) = \frac{\partial{f}}{\partial{x}}i + \frac{\partial{f}}{\partial{y}}j[/tex]

is the vector field. so in your example:

[tex]\frac{\partial{f}}{\partial{x}} = -3[/tex]

[tex]\frac{\partial{f}}{\partial{y}} = 1[/tex]

so:

[tex]grad(y-3x-2) = -3i + j[/tex]

notice that the component of the gradient of your function are constant...thats because your function just a line. if your function were something more complicated, then your components would be functions and your would evaluate them at a particular point because your gradient would then vary as a true vector field.
 
  • #3
i am puzzled. you have not said what the function is, so i do not know what the rgadient is.

if the function is f(x,y) = y-3x = -3x+y, then the gradient is the same everywhere, namely (-3,1).

same if the function is f(x,y) = -3x+y-2 as has been assumed above, but this is not clear from your question. an equation is not a function, unless meant sas the graph of the function, in which case you would be giving the function y = 2-3x whose "gradient is -3.
 
  • #4
I think he just wanted the gradient of a straight line..
 
  • #5
mathwonk said:
i am puzzled. you have not said what the function is, so i do not know what the rgadient is.

if the function is f(x,y) = y-3x = -3x+y, then the gradient is the same everywhere, namely (-3,1).

same if the function is f(x,y) = -3x+y-2 as has been assumed above, but this is not clear from your question. an equation is not a function, unless meant sas the graph of the function, in which case you would be giving the function y = 2-3x whose "gradient is -3.

The gradient vector isn't perpendicular to the slope?
 
  • #6
whozum said:
The gradient vector isn't perpendicular to the slope?

No, for a linear function the gradient is the slope (in "British-speaking" places).
 
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Related to Is the Gradient the Same as the Slope in Linear Functions?

What is a gradient?

A gradient is a visual effect that smoothly transitions between two or more colors, often creating a sense of depth and dimensionality.

How is a gradient created?

A gradient is created by blending two or more colors together, usually in a gradual or seamless manner. This can be done digitally using design software or physically through painting or other artistic techniques.

What is the purpose of using a gradient?

Gradients are often used to add visual interest and depth to designs, as well as to create a sense of movement or direction. They can also be used to convey emotions, such as warmth or coolness, depending on the colors used.

What are the different types of gradients?

There are several types of gradients, including linear, radial, angular, and diamond. Each type creates a different visual effect and is used for different purposes in design.

How do I incorporate gradients into my designs?

Gradients can be incorporated into designs in various ways, such as background colors, overlays, or as part of a specific element or object. Experimenting with different types of gradients and color combinations can help achieve the desired effect in a design.

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