What is the value of b if y=x + b an dintersects with parabola y= x^2 + 4 in one point only?

Question

There is a parabola with eq'n y=x^2 + 4 and a line with equation y= x + b. If they intersect only a one point what would be the value of b?

Please show algebraic solution. Thanks!

Answer 1

The solution is b = 15/4 or 3 3/4.

y = x^2 + 4
y = x + b

x^2 + 4 = x + b
x^2 - x + 4 - b = 0

If there is to be just 1 solution to the above equation, the above equation has to factor to be a perfect square or (x + a)^2 = 0.

So let's complete the square:

We know it has to be (x - 1/2)^2 because of the coefficient of the x term is -1. We divide this by 2 to get -1/2.

so
(x - 1/2)^2 = x^2 - x + 4 - b
x^2 - x + 1/4 - X^2 - x + 4 - b
1/4 = 4 - b
- 15/4 = - b

b = 15/4

and the point where they intersect is (1/2, 4 1/4)

There is your answer!

Note to the above poster: Although the answer you arrived at is the same as mine, I do not think the reasoning is sound. Two equations with the same slope does not necesarrily need to intersect, case in point (two parallel lines). Assumption that the intersecting point will have the same slope is wrong. It just happens to be that for a line to intersect a parabolic line at just one point, the line also happens to be tangent line to the point. It only works if you state this as part of your solution.

Answer 2

The slope of y = x + b is 1

What is the only point on the parabola at which m = 1? (That way, the line can be tangential to the parabola.) You have to differentiate:

y' = 2x; where does y' = 1? at x = 1/2

What is the y value of the parabola at x = 1/2? y = (1/2)^2 + 4 = 1/4 + 4 = 4.25

So (0.5, 4.25) is a point on the line y = x + b

What does b equal? Substitute:
4.25 = .5 + b
3.75 = b

Answer 3

The line has slope 1. The parabola has slope 1 when its derivative = 1, that is, 2x=1. So x = 1/2.
y = (1/2)^2 + 4 = 17/4.
Then Substitute back into y = x + b.
17/4 = 1/2 + b.
15/4 = b.

Answer 4

Given the parabola y = x² - 6x + 4 b) locate the turning element. The turning element is likewise wide-spread because the vertex. allow's positioned the equation into vertex variety through ending up the sq.. y = x² - 6x + 4 y = (x² - 6x + 9) - 5 y - 5 = (x - 3)² The vertex or turning element is (3,5). a) locate axis of symmetry because the squared time period is x the parabola is vertical so the axis of symmetry is vertical also and runs through the vertex. The axis of symmetry is: x = 3 ___________________ 2) Create a quadratic equation with roots {a million/2,-2}. y = (2x - a million)(x + 2) = 2x² + 3x - 2

Answer 5

Draw a picture and convince yourself that the only way these two graphs can intersect in one point only is if they are tangent to one another. Recall that y=x+b always has slope 1, no matter what value of b you use. Find where the parabola has slope 1 (it is at the point (1/2,9/2).) Find your value for b.

Answer 6

y = x + b
y =x^2 +4

x + b = x^2 +4
x^2 -x +(4 -b)

You want to use the quadratic equation to solve for x. There will be only one value of x when the discriminate is = 0

A, B,C = coef of quadratic equation
b = y-intercept
discriminate = B^2 -4AC = 0
(-1)^2 -4(1)(4-b) =0
1 -4(4-b) = 0 now solve for b
1 -16 +4b =0
4b = 15
b = 15/4 = 3.75