Solve p(x)= 2x^4-3x^3-6x^2+5x+6
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Answer:
Step-by-step explanation:
So, we need to find the equation of a parabolic function that goes through the points (-3,67) and (-1,1) and has a stretch factor of 9.
In other words, we want to find a quadratic with a vertical stretch of 9 that goes through the points (-3,67) and (-1,1).
To do so, we first need to write some equations. Let's use the vertex form of the quadratic equation. The vertex form is:
Where a is the leading coefficient and (h,k) is the vertex.
Since a is the leading coefficient, it's also our stretch factor. Thus, let a equal 9.
Also, we have two points. We can interpret them as functions. In other words, (-3,67) means that f(-3) equals 67 and (-1,1) means that f(-1) equals 1. Write the two equations:
And:
Now, we essentially have a system of equations. Thus, to find the original equation, we just need to solve for the vertex. To do so, first isolate the k term in the second equation:
Now, substitute this value to the first equation:
And now, we just have to simplify.
First, from each of the square, factor out a negative 1:
Power of a product property:
The square of -1 is positive 1. Thus, we can ignore them:
Square them. Use the trinomial pattern:
Distribute:
Combine like terms:
The first set cancels. Simplify the second and third:
Subtract 73 from both sides. The right cancels:
Divide both sides by 36:
Therefore, h is -1/6.
Now, plug this back into the equation we isolated to solve for k:
First, remove the negative by simplifying:
Plug in -1/6 for h:
Add. Make 1 into 6/6:
Square:
Multiply. Note that 36 is 9 times 4:
Convert 1 into 4/4 and subtract:
So, the vertex is (-1/6, -21/4).
Now, plug everything back into the very original equation with 9 as a:
And this is our answer :)