Y=(x^2) +1 and y = - (x^2)? tangent

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In summary, to find the equations of the tangent lines to both curves y=(x^2)+1 and y=-(x^2), we must set the slopes of the tangents equal to each other and solve for x0, which represents the x-coordinate of the points where the lines are tangent to the curves. This will give us two symmetric solutions for the equations of the tangent lines, which can be written as y=mx+b where m is equal to the derivative of the functions at the given points and b is determined by solving for x0.
  • #1
swechan02
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Find the equations of the lines that are tangent to both curves simultaneously:y=(x^2) +1 and y = - (x^2)? :eek:
 
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  • #2
Find the equations for the tangent lines to both curves and set them equal to each other. You will find when the slope of the tangents are equal and then can make an equation(s) of of it.
 
  • #3
Jameson is correct but it's a bit more complicated than he implies.

Suppose a line is tangent to y= x2+ 1 at (x0,x02+ 1) and tangent to y= -x2 at (x1,-x12).
Any (non-vertical) line can be written as y= mx+ b. m is equal to the derivative of the functions at the given points: m= 2x0= -2x1 so x1= -x0. We must have x02+1= (2x0)x0+ 1 or b= 1- x02. We must also have -x12= (-2x1)x1+ b or b= x12. That is, b= x12= 1- x02. But since x1= -x0, x12= x02 so 1- x02= x02.

Solve that for x0 and then you can find m and b.

Because of the squares, there are, of course, two symmetric solutions,.
 
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1. What is the equation for y = x^2 + 1?

The equation y = x^2 + 1 is a quadratic equation where the y-value is equal to the square of the x-value plus 1. This equation creates a parabola when graphed.

2. How do you graph y = x^2 + 1?

To graph y = x^2 + 1, you can create a table of values by choosing different x-values and plugging them into the equation to find the corresponding y-values. Then, plot the points on a coordinate plane and connect them to create a parabola.

3. What is the equation for y = -x^2?

The equation y = -x^2 is a quadratic equation where the y-value is equal to the negative square of the x-value. This equation also creates a parabola when graphed, but it is reflected over the x-axis compared to y = x^2.

4. How do you graph y = -x^2?

To graph y = -x^2, you can follow the same steps as graphing y = x^2 + 1, but the parabola will be reflected over the x-axis. You can also use the symmetry of the graph to plot points on one side and then reflect them over the x-axis to complete the graph.

5. What is the relationship between y = x^2 + 1 and y = -x^2?

The two equations y = x^2 + 1 and y = -x^2 are related because they are both quadratic equations that create parabolas when graphed. However, they are reflections of each other over the x-axis. Additionally, y = x^2 + 1 has a positive leading coefficient, which means the parabola opens upwards, while y = -x^2 has a negative leading coefficient, which means the parabola opens downwards.

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