Question

At what points are the equations y=x^2 and y=1/x+2 equal?

Answer 1

The two graphs intersect where the x and y values are exactly the same.  The trouble here is that by setting them equal to each other and solving for x only gives you 1 of the 3 points where those graphs intersect.  The other 2 have to be found on your calculator.  The only point you can solve for analytically is done this way: $x^{2} = \frac{1}{x+2}$.  Multiply both sides by x+2 to get $x^{2} (x+2)=1$.  We then have that $x^{2} =1$ and x+2=1.  Solving the first one for x gives us that x = 1, -1 and the second one gives us that x = -1.  Looking at the graph, we can see that there is no point of intersection at x = 1, but there is in fact one at x = -1.  So that's  1 point of intersection.  The other 2 are found on your calculator.  Graph both the functions in the same viewing window, then hit 2nd and trace and scroll down to 5 for intersection.  You will move your cursor along the first curve to where it appears the 2 intersect and hit enter, then you will be prompted to move the cursor along the second curve to where that cursor overlaps the first.  Hit enter again and then it will display "guess?" and hitting enter again gives you the exact point of intersection.  Besides the (-1,1), our 2 other points of intersection here are (-1.618034, 2.618034) and (.61803399, .38196601).  No way you could tell that from merely looking at the graph!!! Let your calculator work for you...it can do AMAZING things!!!!!! Learn as much about it as you can!!

Answer 2

Answer:

The points where equations are equal are (-1,1),(-0.618,0.382), (1.618,2.618).

Step-by-step explanation:

Given : Equations  $y=x^2$ and  $y=\frac{1}{x+2}$

To find :  At what points are the equations are equal?

Solution :

Equations  $y=x^2$ and  $y=\frac{1}{x+2}$

The intersection points of both graphs are the points where both equations have the same value .

So, The points at which the equations are equal through given graph is at B,C, F.

We determine the points by plotting in graphing tool.

Point B is  (1.618,2.618)

Point C is (-1,1)

Point F is (-0.618,0.382)

Refer the attached figure below.

Therefore, The points where equations are equal are (-1,1),(-0.618,0.382), (1.618,2.618).