The new function PQ is called the quadratic function, and it is defined as PQ(x) = F(x)g(x) + Q(x), where the + means we include both F(x) and Q(x) in the new function.
Quadratic functions are functions where the output value is dependent on two variables, x and x2. The output value is determined by both the input value and how much the input value changes in magnitude.
The graph of a quadratic function shows two points of inflection, or points where the curve changes its direction or slope. The curve will either change direction (a minimum or a maximum) or slope (rising or falling) at these points.
This article will discuss these topics in detail.
Calculate P(-1) and Q(-1)
P(-1) = F(-1)g(-1) = |-2|
Q(-1) = F(-1)/G(-1) = -|2|/2
Because P(-) is a function, you can’t just plug in -1 for x, so you have to find the inverse function. The inverse of P(x) is P(-x). You can find this by finding when x = -P(-x), which is when -P(x) = 0.
Therefore, the inverse of P(x) is P(-x)=¬P(x)=¬|-2|=|-2|. Plugging this in we get: Q(-1)=¬-
Because Q(-) isn’t a function, you have to find the value of Q(-1)=¬
This can be tricky because there are no clear rules for finding the zeroes of quotients.
Quotients can be solved using logarithms or fractional exponents. Logarithms will be discussed here.
- Find the logarithm base b of Q(-) using the formula logbQ=logbF/logbG.
- Then, solve for Q by putting in zeros for F and G.
- If Q=-
Q
- If Q=-
For more information on quotients visit this website.
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Calculate P(-2) and Q(-2)
The next step is to calculate the derivative of the Constant function. The Constant function has no variable, so its derivative is simply 0.
The derivative of P(-2) is 0 – because P(-2) = F(-2)g(-2), and F(-)=0, then P(-2)=0. The same applies to Q(-2) = 0.
Next, calculate the derivative of Q(-2). Since Q(-)=F(-)/G(-) and G(-)=0, then Q(-2)=F'(-2)=0. This checks out! The graph does not change when -2 is substituted for x, so there is no y’’-axis crossing point at -2.
Now that we have proven our points, let’s look at some real-world examples using this differentiation technique.
These examples will be based on real life situations and how they can be solved using this method. Let’s get started!%20Read%20More%20<%20/em%20><%20/strong%20>%20)%20example%201:%20A%20car%20slows%20down%20as%20it%20approaches%20a%20stoplight.
This%20could%20be%20modeled%20by%20solving%20for%20the%20speed%20of%20the%20car%20as%20a%20function%20of%20time%20until%20it%20stops%20at%20the%20light.
<%20br%20>To%20model%20this%20situation,%20we%20need%20to%20know%20two%20things:%20how%20fast%20the%20car%20moves%20(speed),%20and%20how%20long%20it%20takes%20to%20stop%20(time).%20We%20will%20assume%20that%20it%20takes%20one%20full%20minute%20for%20the%20car%20to%20stop.<%20br%20><%20br%20>Let’s%20start%20by%20solving%20for%20V(t),%20which%20is%20the%20speed%20of%20the%20car%20as%20a%20function%20of%20time%20t%20(time%20until%20it%20stops).<%20br%20><%20br%20>We%20know%20that%20V(t)=0,%20so%20our%20equation%20becomes%20V=constant*t%20where%20constant=1.<%20br%20><%20br%20>Now%20we%20need%20to%20find%20what%20V%20equals-%20we%20do%20this%20by%20taking%20its%20derivative%20with%20respect%20to%20t%20(DV/TDt).
Conclude%20the%20relationship%20between%20the%20functions
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%20By%20definition,%20the%20power%20function%20P(x)%20equals%20F(x)g(x)%20times%20Q(x),%20where%20Q%20is%20the%20constant%20function%201.
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Therefore,%20P(x)%20=%20F(x)g(x)*1%20=%20F(x).%20By%20simplifying%20the%20left%20side%20of%20this%20equation,%20we%20can%20conclude%20that%20P%20is%20equal%20to%20F.
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By%20using%20a%20similar%20method,%20you%20can%20show%20that%20Q(x)%20=%20G(x)*1%20=%20G.%20You%20can%20also%20show%20that%20P=Q%20by%20simplifying%20the%20left%20side%20of%20this%20equation.
Graph%20the%20functions%20and%20observe%20their%20relationship
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%20In%20this%20case,%20the%20function%20P%20is%20a%20function%20whose%20graph%20is%20shown,%20and%20the%20functions%20F%20and%20G%20are%20derived%20functions%20whose%20graphs%20are%20not%20shown.%20The%20same%20is%20true%20for%20Q.
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By%20graphing%20P(x)%20=%20F(x)g(x)%20and%20Q(x)%20=%20F(x)/G(x),%20Where%20F%20and%20G%20Are%20the%20Functions%20Whose%20Graphs%20Are%20Shown,%20you%20can%20see%20how%20they%20relate%20to%20each%20other.
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For%20example,%20if%20P(x)%20=%200%20then%20either%20F(x)%20=%200%20or%20g(x)%20=%200.%20If%20P=0%20then%20either%20F=0%20or%20G=0.%20Since%20neither%20of%20these%20are%20true,%20then%20P%20cannot%20be%20equal%20to%200%20which%20means%20that%20at%20least%20one%20of%20them%20must%20be%20greater%20than%200.
