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Given a tabular function, create a new row to represent a vertical shift.

  1. Identify the output row or column.
  2. Determine the magnitude of the shift.
  3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.

Shifting a tabular function vertically

A function f ( x ) is given in [link] . Create a table for the function g ( x ) = f ( x ) 3.

x 2 4 6 8
f ( x ) 1 3 7 11

The formula g ( x ) = f ( x ) 3 tells us that we can find the output values of g by subtracting 3 from the output values of f . For example:

f ( 2 ) = 1 Given g ( x ) = f ( x ) 3 Given transformation g ( 2 ) = f ( 2 ) 3 = 1 3 = 2

Subtracting 3 from each f ( x ) value, we can complete a table of values for g ( x ) as shown in [link] .

x 2 4 6 8
f ( x ) 1 3 7 11
g ( x ) −2 0 4 8
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The function h ( t ) = 4.9 t 2 + 30 t gives the height h of a ball (in meters) thrown upward from the ground after t seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function b ( t ) to h ( t ) , and then find a formula for b ( t ) .

b ( t ) = h ( t ) + 10 = 4.9 t 2 + 30 t + 10
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Identifying horizontal shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift , shown in [link] .

Horizontal shift of the function f ( x ) = x 3 . Note that h = + 1 shifts the graph to the left, that is, towards negative values of x .

For example, if f ( x ) = x 2 , then g ( x ) = ( x 2 ) 2 is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in f .

Horizontal shift

Given a function f , a new function g ( x ) = f ( x h ) , where h is a constant, is a horizontal shift    of the function f . If h is positive, the graph will shift right. If h is negative, the graph will shift left.

Adding a constant to an input

Returning to our building airflow example from [link] , suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.

We can set V ( t ) to be the original program and F ( t ) to be the revised program.

V ( t ) =  the original venting plan F ( t ) = starting 2 hrs sooner

In the new graph, at each time, the airflow is the same as the original function V was 2 hours later. For example, in the original function V , the airflow starts to change at 8 a.m., whereas for the function F , the airflow starts to change at 6 a.m. The comparable function values are V ( 8 ) = F ( 6 ) . See [link] . Notice also that the vents first opened to 220  ft 2 at 10 a.m. under the original plan, while under the new plan the vents reach 220  ft 2 at 8 a.m., so V ( 10 ) = F ( 8 ) .

In both cases, we see that, because F ( t ) starts 2 hours sooner, h = 2. That means that the same output values are reached when F ( t ) = V ( t ( 2 ) ) = V ( t + 2 ) .

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Questions & Answers

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rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
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if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
What is domain
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
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This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
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Confunction Identity
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For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
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It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
I would like to add that they are used in AC signal analysis for one thing
Good call Scott. Also radar signals I believe.
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations

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