A quadratic function is defined by f(x)=-3.7x^2+6.8x+4.2. A linear function is defined by g(x)=-0.5x+k. Determine the value of k so that the line intersects the parabola at exactly one point. Write your answer to the nearest hundredth
one point? They must be tangent.
The slope of the quadratic is
f'=slope=-7.4x+6.8
the slope of g is -.5
setting them equal, then
-.5=-7.4x+6.8
solve for x.
To find the value of k that makes the linear function g(x) intersect the quadratic function f(x) at exactly one point, we need to determine the discriminant of the quadratic equation.
The discriminant, denoted as Δ, is the part of the quadratic formula that helps us determine the number of solutions to the equation. If Δ is greater than zero, there are two distinct solutions (the parabola crosses the x-axis at two points). If Δ is equal to zero, there is exactly one solution (the parabola touches the x-axis at one point). If Δ is less than zero, there are no real solutions (the parabola does not intersect the x-axis).
The quadratic function f(x) is defined as f(x) = -3.7x^2 + 6.8x + 4.2. To find the discriminant, we can use the quadratic formula:
Δ = b^2 - 4ac
where a, b, and c are coefficients of the quadratic function f(x) in the form of ax^2 + bx + c.
In this case, a = -3.7, b = 6.8, and c = 4.2. Plugging these values into the formula, we get:
Δ = (6.8)^2 - 4(-3.7)(4.2)
Δ = 46.24 + 62.16
Δ = 108.40
Since the discriminant is positive (Δ > 0), the quadratic equation has two distinct solutions, and therefore, the quadratic function f(x) intersects the x-axis at two points.
To have exactly one intersection point between g(x) and f(x), the linear function g(x) must touch the parabola at exactly one point. This means that the discriminant of the quadratic equation should be equal to zero (Δ = 0).
We know the linear function g(x) is defined as g(x) = -0.5x + k. So we can find the discriminant for g(x) as well.
Δ = b^2 - 4ac
where a = 0, b = -0.5, and c = k.
Plugging these values into the formula, we get:
Δ = (-0.5)^2 - 4(0)(k)
Δ = 0.25
Since Δ = 0.25, we have a positive value, which means the linear function g(x) and the quadratic function f(x) will intersect at exactly one point if the discriminant is equal to zero.
Setting Δ equal to zero, we have:
0.25 = 0
This equation has no solutions. Therefore, there is no value of k for which the line g(x) = -0.5x + k intersects the parabola f(x) = -3.7x^2 + 6.8x + 4.2 at exactly one point.