For what value of b will the line y=-2x+b be tangent to the parabola y=3x^2+4x-1 ? (Show your work - 2 marks)
I love it when you guys as us to show our work. Ever think of showing some of yours?
y' = 6x+4
The line given has slope -2.
So, when do we have
6x+4 = -2?
x = -1
y(-1) = -2, so we want the point (1,-2) to be on the line. That means
-2 = -2(-1)+b
b = -4
See the graphs at
http://www.wolframalpha.com/input/?i=plot+y%3D3x^2%2B4x-1%2C+y%3D-2x-4
To find the value of b for which the line y = -2x + b is tangent to the parabola y = 3x^2 + 4x - 1, we need to set the equations equal to each other and solve for x and y.
Setting the equations equal to each other:
-2x + b = 3x^2 + 4x - 1
Rearranging the equation:
3x^2 + 6x + (b + 1) = 0
For the line to be tangent to the parabola, the quadratic equation above must have a single root, known as a double root. This means the discriminant (b^2 - 4ac) of the quadratic equation must be equal to zero.
Using the quadratic formula where a = 3, b = 6, and c = b + 1, the discriminant is given by:
Discriminant = (6)^2 - 4(3)(b + 1)
= 36 - 12(b + 1)
= 36 - 12b - 12
= 24 - 12b
Setting the discriminant equal to zero and solving for b:
24 - 12b = 0
-12b = -24
b = -24 / -12
b = 2
Therefore, the value of b for which the line y = -2x + b is tangent to the parabola y = 3x^2 + 4x - 1 is b = 2.
To find the value of b for which the line y=-2x+b is tangent to the parabola y=3x^2+4x-1, we need to find the point of intersection between the line and the parabola.
Step 1: Set the equations equal to each other:
-2x + b = 3x^2 + 4x - 1
Step 2: Simplify the equation:
3x^2 + 6x + (b + 1) = 0
Step 3: Use the discriminant to determine when the line is tangent to the parabola. Since the line is tangent, the discriminant must be equal to 0.
The discriminant is given by: b^2 - 4ac = 0
In our case, a = 3, b = 6, and c = b + 1.
Substituting these values into the discriminant equation:
(6)^2 - 4(3)(b + 1) = 0
Step 4: Solve the equation for b:
36 - 12(b + 1) = 0
36 - 12b - 12 = 0
24 - 12b = 0
-12b = -24
b = -24 / -12
b = 2
Therefore, the value of b for which the line y=-2x+b is tangent to the parabola y=3x^2+4x-1 is b = 2.