I have 2 question

1.The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$?

2.What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?

Guest Feb 22, 2018

#1**+2 **

I have 2 question

1.The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$?

2.What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?

1. The vertex (3, 5) and the point (0, - 13) are on the graph ....and (3,5) is tthe vertex

The x coordinate of the vertex is given by :

-b / [2a] = 3

-b = 6a

b = - 6a

And since (0, - 13) is on the graph, then c = -13

And using the vertex, we can find "a" thusly

5 = a(3)^2 - 6a(3) - 13

18 = 9a- 18a

18 = -9a

-2 = a ⇒ b = -6(-2) = 12

So...our function is

y = -2x^2 + 12x - 13

And when x = 4, m =

-2(4)^2 + 12(4) - 13

-32 + 48 - 13

3

So....the point (4,3) is on the graph

Here's a graph : https://www.desmos.com/calculator/jhmml2xcki

CPhill
Feb 22, 2018

#2**+2 **

2.What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?

Rewrite as

x^2 + 14x +y^2 - 4y = 10

This is a circle....we need to complete the square on x and y [ I'm assuming that you are familiar with this?? ]

x^2 + 14x + 49 + y^2 - 4y + 4 = 10 + 49 + 4

The right side simplifies to 63 = ( radius of the circle)^2

So... the area is pi * r^2 = pi *63 ≈ 197.9 units^2

CPhill
Feb 22, 2018