Derivações e transformações pelo metodo da tangente

(Videos) Introduction to Function Transformations, Horizontal and Vertical Transformations, and Function Transformations

by MySecretMathTutor, PatrickJMT, and Mathispower4u

What is a Transformation?

The transformations that functions under go have an effect on the tangent lines. Considering this, there are different types of transformations that have different effects.

Types:

Vertical Translation is when a function moves upwards or downwards by a certain amount. As a result of this transformation, the tangent line moves along with the function. For instance, the original function of a graph is $y=x^2+2$ at $x=1$ and this function underwent a vertical translation of +3. So what would the tangent line be after the transformations if the original tangent line function is $y=2x+1$. It would be $y=2x+4 because the graph moved up, so the y-intercept would change. In this case, verticals translations do not affect the slope and the derivatives stays the same.

Here is the function $g(x)=x^2-1$. Draw the tangent line at $x=1$.

Horizontal Translation is when a function moves to the right or to the left by the same amount. So with the function, the tangent lines also moves to the left or right. This transformation changes the derivative function because the X value changes. For example, when the original function is $y=(x+1)^2$, the tangent line would be $y=2x+2$. Here, one notices that the slope of the tangent is 2 which affects the x values.

Stretching is when a function is stretched or compressed by multiplying the entire function by the same factor. In this case, the derivative is affected. The slope of the function decreases by a certain amount and the slope of the tangent line also decreases by certain amount too.

Reflection over x-axis is when the function of a graph reflected over the x-axis by changing the sign of the domain. With this transformation, the sign of the derivative also changes.

Effects on transformation of a tangent- Whatever transformation a function undergoes, the tangent lines to the function also undergoes the exact same transformation as the function. For example;If the function was to slide up 3 units the tangent line will also move up three units.As the Tangent line(s) follows the function, when the function increase or decrease, the Tangent line's slope remains the same. It does not change. The derivative of a transformed function is equal to the transformation of the tangent line and derivative equations.The stretch and reflection transformations affect the slope, but vertical shifts do not.

As shown on this graph, even though the function changes, the slope of the tangent line does not chnage.

Efects of transformations on Derivatives- When translated and reflected,the derivative opf a function will always be the same.How ever this is not true when the function is Streched/dialated.It will not be the same in this caseThe ways in which specific transformations affect the shape and location of a function's graph are the same no matter the type of function, as are the ways in which transformations affect the derivatives and tangent lines of a function. For example, the derivative of the quadratic function, f(x)=x^2 is f'(x)=2x. When this function undergoes the transformation of a vertical stretch by a factor of 2 (f(x)=2x^2), the derivative is also multiplied by 2 (f'(x)=2x -> f'(x)=4x). This rule applies to all types of functions.

Effects on transformation of a tangent line as shown on this graph, even though the function changes, the slope of the tangent line does not change. Whenever a transformation function undergoes, the tangent lines to the function also undergoes the exact same transformation as the function. For example: If the function moved up 3 units, the tangent line will also move up 3 units. As the Tangent line(s) follows the function, when the function increases or decreases, the tangent line's slope remains the same and it does not change. The derivative of a transformed function is equal to the transformation of the tangent line and derivative equations.The stretch and reflection transformations affect the slope, but vertical shifts do not.

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