+0

0
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Find the vertex of the graph of the equation $x - y^2 + 8y = 13.$

Find the vertex of the graph of the equation $y = -2x^2 + 8x - 15.$

Find the area of the region enclosed by the graph of $x^2 + y^2 = 2x - 6y + 6$

The graph of $y = ax^2 + bx + c$ has an axis of symmetry of $x = 4.$ Find $\frac{b}{a}$.

The graph of $y = ax^2 + bx + c$ is shown below. Find $a \cdot b \cdot c$.  (The distance between the grid lines is one unit.)

May 28, 2021

#1
+102
0

@Batterydude6000, I can tell you are taking an AoPS class, so I won't be sharing the answers but I'll give hints.

1 - We transform the equation into $$x = y^2 - 8y + 13$$. We can then complete the square in $$y$$, which turns the equation into $$x = (y-4)^2 - 3$$. See if you can solve yourself from here.

2 - Again, we can complete the square to get $$y = -2(x^2-4x)-15$$. This simplifies to $$y = -2(x - 2)^2 - 7$$. Try solving it from here.

3 - To complete the square, we can add $$6y$$ and subtract $$2x$$, then add $$1$$ to complete the square for $$x$$ and add $$9$$ to complete the square for $$y$$. It should be pretty simple from here.

4 - Since the axis of symmetry is $$x = 4$$, we can substitute that into $$y = a(x-h)^2+k$$

5 - The simplified equation of $$y = a(x-h)^2+k$$ is $$y = a(x + 3)^2 - 2$$. We can also tell that the parabola passes through the point $$(-1,0)$$, so $$x = -1, ~ y = 0$$. Try substituting that in the equation and sees where that gets you.

Hope that helped.

AnxiousLlama

May 28, 2021
#3
+36434
+2

1.)  x = y^2 -8y + 13

re-arrange to vertex form

x = (y-4)^2  -3        vertex is   -3,4

May 28, 2021
#4
+36434
+2

2 )   -2x^2+8x-15     re-arrange to vertex form

-2 (x^2 - 4x) -15

-2 ( x-2)^2  - 7         vertex = 2, -7

May 28, 2021
#5
+36434
+2

3)  x^2 -2x    + y^2  + 6y   =  6     Arrange into standard circle equation form

(x-1)^2      +      ( y+3)^2 = 6 +1 + 9

6+ 1 + 9 = 16  = r^2

Area of a circle = pi r^2 = pi (16)  = 16 pi  units2

May 28, 2021
#6
+27
0

Can you answer the last two questions? Thank you so much for the other answers.... all correct!

Jun 5, 2021