3 The The curve y=ax² +bac + 5 where a and b are constants has a turning point p (1,3). find the values of a and b and determine
1 answer:
Answer:
a = 2, b = -2
P is at a minimum
Step-by-step explanation:
For any polynomial function, the turning point is either a maximum or a minimum
It can be determined by taking the first derivative of the function and setting it equal to 0 and solving for x and y
In this case we are given the turning point as x=1, y = 3 and we have to calculate a and b in the equation y = ax² + (ab)x + 5
The first derivative of y, y' = 2ax + ab
If we set this equal to 0 we get 2ax + ab = 0 ==> 2ax = -ab ==> b = -2
So the equation is of the form y = ax² -2ax + 5 (subbing for b)
Since we know that at x = 1, y =3 substitute y and x values in the above equation and solve for a
y = 3 = a(1²) - 2a(1) + 5
a - 2a + 5 = 3
-a + 5 = 3
a = 2
So the equation is of the form y = 2x² -4x + 5
If we plot this we will find that P is a minimum point
However, we can always determine mathematically if P is a max or min by taking the second derivative of the original function and noting the sign. If it is positive, the point is a minimum, and if it is negative, the point is a maximum.
Taking derivatives,
y' = 4x - 4
y'' = (4x-4)' = 4
The sign is positive so P is a minimum
Graph attached for reference
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General Idea:
Domain of a function means the values of x which will give a DEFINED output for the function.
Applying the concept:
Given that the x represent the time in seconds, f(x) represent the height of food packet.
Time cannot be a negative value, so
The height of the food packet cannot be a negative value, so
We need to replace for f(x) in the above inequality to find the domain.
The possible solutions of the above inequality are given by the intervals . We need to pick test point from each possible solution interval and check whether that test point make the inequality
true. Only the test point from the solution interval [-200, 200] make the inequality true.
The values of x which will make the above inequality TRUE is
But we already know x should be positive, because time cannot be negative.
Conclusion:
Domain of the given function is
Answer:
Value of
Step-by-step explanation:
Given: A vertical line is dropped from the x-axis to the point (12, -9) as shown in the diagram below;
To find the value of .
By definition of secants;
Now, first find the cosine of angle
As the point (12 , -9) lies in the IV quadrant , where
Consider a right angle triangle;
here, Adjacent side = 12 units and Opposite side = -9 units
Using Pythagoras theorem;
or
Cosine ratio is defined as in a right angle triangle, the ratio of adjacent side to hypotenuse side.
then;
and
therefore, the value of is,
