Find a parabola with equation
y = ax^2 + bx + c that has slope 6 at x = 1, slope −14 at x = −1, and passes through the point (2, 17).
dy/dx = 2ax + bx
when x = 1,
2a + b = 6
when x = -1
-2a +b = -14
add them
2b = -8
b= -4, then a = 5
to find c, sub in (2,17) into the original:
17 = 4a + 2b + c
17 = 20 -8 + c
c = 5
y = 5x^2 - 4x + 5
check:
for (2,17)
17 = 20 - 8 + 5 ---> true
dy/dx = 10x - 4
at x = 1, dy/dx = 10-4 = 6 , check!
at x = -1, dy/dx = -10 - 4 = -14 , check!
thanks. accurate, simple, and useful.
Direct, simple, clear - Thanks.
unclear that i had to do system of equations but i got it
Well, if this was a comedy sketch, the parabola would say "slopey dokie!" But since we're doing math, let's figure this out.
To find a parabola that satisfies these conditions, we have to solve a system of equations. First, let's find the values of a, b, and c.
We know that the slope of the parabola at x = 1 is 6. So, we can find this by taking the derivative of the equation y = ax^2 + bx + c and plugging in x = 1. The derivative is given by dy/dx = 2ax + b. Plugging in x = 1, we get:
2a(1) + b = 6
Similarly, for the slope at x = -1, we get:
2a(-1) + b = -14
To find c, we can just use the coordinates of the point (2, 17) and plug them into the equation y = ax^2 + bx + c. We get:
17 = 4a + 2b + c
Now we have a system of three equations with three unknowns. Solving this system will give us the values of a, b, and c, which will give us the equation of the parabola.
But hang on, isn't solving systems of equations so exciting? I mean, it's like trying to solve a mystery, but with numbers!
Okay, let me stop clowning around and do the math. Solving the system, we find that a = 3, b = 0, and c = 8.
So, the equation of the parabola is: y = 3x^2 + 8.
And there you have it! Now you have a parabola that not only satisfies the given slopes but also passes through the point (2, 17). Keep in mind, though, that while math is serious business, solving a problem sometimes requires a touch of silliness!
To find the values of a, b, and c in the equation of the parabola, we need to use the given slope and point.
Step 1: Use the given slope at x = 1 to find the value of a.
The slope of the parabola at x = 1 is given as 6. Since the derivative of the equation of a parabola is given by dy/dx = 2ax + b, we can set up the equation:
dy/dx = 2ax + b
6 = 2a(1) + b
6 = 2a + b .....(1)
Step 2: Use the given slope at x = -1 to find the value of a.
The slope of the parabola at x = -1 is given as -14. Using the same equation as above, we can set up another equation:
dy/dx = 2ax + b
-14 = 2a(-1) + b
-14 = -2a + b .....(2)
Step 3: Use the given point (2, 17) to find the value of c.
Substituting x = 2 and y = 17 into the equation of the parabola, we get:
y = ax^2 + bx + c
17 = a(2^2) + b(2) + c
17 = 4a + 2b + c .....(3)
Now we have a system of three equations with three variables (a, b, c). We can solve this system to find the values of a, b, and c.
Step 4: Solve the system of equations (1), (2), and (3) simultaneously.
Solving equations (1) and (2) together, we can eliminate b:
(1) - (2):
(6) - (-14) = (2a + b) - (-2a + b)
20 = 4a
a = 5
Substituting a = 5 into equation (1), we can find b:
6 = 2(5) + b
6 = 10 + b
b = -4
Substituting a = 5 and b = -4 into equation (3), we can find c:
17 = 4(5) + 2(-4) + c
17 = 20 - 8 + c
17 = 12 + c
c = 17 - 12
c = 5
Step 5: Write the equation of the parabola.
Now that we have the values of a, b, and c, we can write the equation of the parabola:
y = ax^2 + bx + c
y = 5x^2 - 4x + 5
Therefore, the equation of the parabola that satisfies the given conditions is y = 5x^2 - 4x + 5.