Answers:
k = -3.5
Intersection point is (0.5, -2.5)
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Step-by-step explanation:
Apply the derivative to y=2x^2-3 and you should get dy/dx = 4x
The derivative helps determine the slope of the tangent at any point on the curve.
The slope of the tangent line y = 2x+k is 2.
We want the slope of the tangent to be 2, so we'll replace the dy/dx with 2 and solve for x.
dy/dx = 4x
2 = 4x
x = 2/4
x = 0.5
Plug this into the curve's original equation.
y = 2x^2 - 3
y = 2(0.5)^2 - 3
y = -2.5
Therefore, the tangent line y = 2x+k and the curve y = 2x^2-3 intersect at the point (0.5, -2.5). This is the point of tangency.
We'll use the coordinates of this point to determine k.
y = 2x+k
-2.5 = 2(0.5) + k
-2.5 = 1 + k
k = -2.5-1
k = -3.5
Visual verification is shown below. I used GeoGebra to make the graph, but you could use any other tool you prefer (such as Desmos).