Note on the Definition of a Lineal Functional
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The zero from solving the linear function above graphically must match solving the same function algebraically. This is the same zero that was found using the graphing method. The slope-intercept form of a line summarizes the information necessary to quickly construct its graph.
One of the most common representations for a line is with the slope-intercept form. This assists in finding solutions to various problems, such as graphing, comparing two lines to determine if they are parallel or perpendicular and solving a system of equations. Simply substitute the values into the slope-intercept form to obtain:. The value of the slope dictates where to place the next point.
Introduction to Linear Functions
Using this information, graphing is easy. The point-slope equation is another way to represent a line; only the slope and a single point are needed. Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation. The point-slope equation is a way of describing the equation of a line. Therefore, the two equations are equivalent and either one can express an equation of a line depending on what information is given in the problem or what type of equation is requested in the problem. Plug this point and the calculated slope into the point-slope equation to get:.
Be careful if one of the coordinates is a negative. Distributing the negative sign through the parentheses, the final equation is:. Again, the two forms of the equations are equivalent to each other and produce the same line. The only difference is the form that they are written in.
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Standard form is another way of arranging a linear equation. In the standard form, a linear equation is written as:.
The graph of the equation is a straight line, and every straight line can be represented by an equation in the standard form. We know that the y -intercept of a linear equation can easily be found by putting the equation in slope-intercept form. However, the zero of the equation is not immediately obvious when the linear equation is in this form.
Skip to main content. Linear Functions. Search for:. Learning Objectives Describe the parts and characteristics of a linear function.
Key Takeaways Key Points A linear function is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable. A function is a relation with the property that each input is related to exactly one output. A relation is a set of ordered pairs. The graph of a linear function is a straight line, but a vertical line is not the graph of a function.
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Key Terms relation : A collection of ordered pairs. Key Takeaways Key Points The slope of a line is a number that describes both the direction and the steepness of the line; its sign indicates the direction, while its magnitude indicates the steepness. Key Terms steepness : The rate at which a function is deviating from a reference. Learning Objectives Recognize examples of functions that vary directly and inversely. Key Takeaways Key Points Two variables that change proportionally to one another are said to be in direct variation. The relationship between two directly proportionate variable can be represented by a linear equation in slope -intercept form, and is easily modeled using a linear graph.
Inverse variation is the opposite of direct variation; two variables are said to be inversely proportional when a change is performed on one variable and the opposite happens to the other. The relationship between two inversely proportionate variables cannot be represented by a linear equation, and its graphical representation is not a line, but a hyperbola. Key Terms hyperbola : A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.
Two magnitudes numbers are said to be proportional if the second varies in a direct relation arithmetically to the first. Learning Objectives Practice finding the zeros of linear functions. Zeros can be observed graphically or solved for algebraically. A linear function can have none, one, or infinitely many zeros. If the line is non-horizontal, it will have one zero.
Learning Objectives Convert linear equations to slope-intercept form and explain why it is useful. Key Terms slope : The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical. Learning Objectives Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation. The point-slope equation requires that there is at least one point and the slope.
If there are two points and no slope, the slope can be calculated from the two points first and then choose one of the two points to write the equation.
The point-slope equation and slope-intercept equations are equivalent. Learning Objectives Explain the process and usefulness of converting linear equations to standard form. This process is known as back-propagation. Activation functions make the back-propagation possible since the gradients are supplied along with the error to update the weights and biases. Why do we need Non-linear activation functions :- A neural network without an activation function is essentially just a linear regression model.
The activation function does the non-linear transformation to the input making it capable to learn and perform more complex tasks. Suppose we have a Neural net like this Neuron can not learn with just a linear function attached to it. A non-linear activation function will let it learn as per the difference w. Hence we need activation function. For example : Calculation of price of a house is a regression problem. Even in this case neural net must have any non-linear function at hidden layers.
Tanh Function :- The activation that works almost always better than sigmoid function is Tanh function also knows as Tangent Hyperbolic function. Both are similar and can be derived from each other. It is the most widely used activation function. Chiefly implemented in hidden layers of Neural network.
Softmax Function :- The softmax function is also a type of sigmoid function but is handy when we are trying to handle classification problems. Foot Note :- The activation function does the non-linear transformation to the input making it capable to learn and perform more complex tasks. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.
See your article appearing on the GeeksforGeeks main page and help other Geeks. Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. Writing code in comment? Please use ide. How to Start Learning Machine Learning? What is an activation function and why to use them? Mathematical proof :- Suppose we have a Neural net like this :- Elements of the diagram :- Hidden layer i.
Note: We are not considering activation function here Layer 2 i.