What are the coordinates of red point?

Question

The function $f (x)=-x^2+4$ "in red" is moving along the line $y=x+4$ " in black " from green point to black point and becomes in the place of blue graph as shown in the following graph

enter image description here

I got the green point $(0,4)$ by using the equation $y=x+4$ , and got the intersection point of the two parabolas $(2,0)$ and I stopped here.

@Brevan Ellefsen i am sorry for making misunderstanding about the post , i edited the post ; hope that will reflect my intention — user373141, Apr 14 '17 at 00:16
Now that all of that is out of the way, what are your thoughts on this problem? What have you tried so far? — amd, Apr 14 '17 at 00:36
@amd i got the green point $(0,4)$ by using the equation $y=x+4$ , and got the intersection point of the two parabolas $(2,0)$ ; and i stopped here — user373141, Apr 14 '17 at 00:41
If we call the unknown $x$-coordinate of the black point $x_c$, what would be the equation of the blue parabola? — amd, Apr 14 '17 at 00:45

Ovi · Accepted Answer · 2017-04-14T01:15:44.883

11

We can solve this problem by remembering function transformations.

The equation of the original parabola is $y = -x^2 + 4$. You are shifting the parabola along a line of slope $1$, which means that the parabola is shifting up the same ammount as it is shifting to the right. So the equation of the blue parabola will be $y = -(x-h)^2 + 4+h$, where $h$ is a positive constant.

Since the blue parabola and the red parabola share an $x$ intercept, we know that $(2, 0)$ satisfies $y = -(x-h)^2 + 4 + h$. If we plug in $(2, 0)$, we can solve for $h$, and then we are done. Can you do the rest?

edited Apr 14 '17 at 01:15

answered Apr 14 '17 at 00:54

Ovi

23,737

2

thank you for help; i got $h=5$ so the vertex is $(5,9)$ , right? – user373141 Apr 14 '17 at 01:03
1

@prayersmith Yes you are correct! – Ovi Apr 14 '17 at 01:09
and so the red point will be $(8,0)$ ; Elegant Solution and reasoning; thank you very much – user373141 Apr 14 '17 at 01:12
1

@prayersmith Correct! You're welcome. – Ovi Apr 14 '17 at 01:14

score 7 · Answer 2 · answered Apr 14 '17 at 00:52

If you slide the green point some distance $d$ to the right along the line, then both the $x$ and $y$ coordinates of the point increase by $d$. This means that the shifted parabola has the equation $(y-d)=4-(x-d)^2$. You know that this shifted parabola passes through the intersection point $(2,0)$, so plug that in and solve for $d$, then find the other $x$-intercept of the resulting equation (you can use symmetry to make that last step really easy).

score 6 · Answer 3 · answered Apr 14 '17 at 01:12

The other answers are good but here's another way to do it. Given a parabola with it's vertex on the line $y=x+4$ and $a=-1$, and a zero at $x=2$, the height of the parabola will be $x+4$, and $x$-distance from the vertex to the zero will be $|x-2|$. Thus, $(x+4)-|x-2|^2 = 0$, or $x^2-5x = 0$, which means that $x=0$ (the red parabola), or $x=5$ (the blue parabola). Since the distance from the vertex to each of it's roots are the same, we calculate $|x-2|$, and we can add and subtract that to $x$, the $x$-coordinate of the vertex. This gives us the two zeros of the blue parabola.

What are the coordinates of red point?

3 Answers3