Answer:
The vertex of f(x) is at (-5,-10).
The function g(x) is decreasing at a rate of 5%.
The zeros of h(x) are at x = -3 and x = 1.
Step-by-step explanation:
Vertex of a quadratic function:
Suppose we have a quadratic function in the following format:
![f(x) = ax^{2} + bx + c](https://tex.z-dn.net/?f=f%28x%29%20%3D%20ax%5E%7B2%7D%20%2B%20bx%20%2B%20c)
It's vertex is the point ![(x_{v}, f(x_{v})](https://tex.z-dn.net/?f=%28x_%7Bv%7D%2C%20f%28x_%7Bv%7D%29)
In which
![x_{v} = -\frac{b}{2a}](https://tex.z-dn.net/?f=x_%7Bv%7D%20%3D%20-%5Cfrac%7Bb%7D%7B2a%7D)
If a<0, the vertex is a maximum point, that is, the maximum value happens at
, and it's value is ![f(x_{v})](https://tex.z-dn.net/?f=f%28x_%7Bv%7D%29)
Decaying exponential function:
A decaying exponential function has the following format:
![A(x) = A(0)(1-r)^{x}](https://tex.z-dn.net/?f=A%28x%29%20%3D%20A%280%29%281-r%29%5E%7Bx%7D)
In which A(0) is the initial quantity and r is the decay rate.
Zeros of a function:
Given a polynomial f(x), this polynomial has roots
such that it can be written as:
, in which a is the leading coefficient.
Vertex of f:
The function f is given by:
![3(x+5)^2 - 10](https://tex.z-dn.net/?f=3%28x%2B5%29%5E2%20-%2010)
Expanding the calculations to place at the correct format:
![3(x+5)^2 - 10 = 3(x^2 + 10x + 25) - 10 = 3x^2 + 30x + 75 - 10 = 3x^2 + 30x + 65](https://tex.z-dn.net/?f=3%28x%2B5%29%5E2%20-%2010%20%3D%203%28x%5E2%20%2B%2010x%20%2B%2025%29%20-%2010%20%3D%203x%5E2%20%2B%2030x%20%2B%2075%20-%2010%20%3D%203x%5E2%20%2B%2030x%20%2B%2065)
Which means that
![a = 3, b = 30, c = 65](https://tex.z-dn.net/?f=a%20%3D%203%2C%20b%20%3D%2030%2C%20c%20%3D%2065)
The x-value of the vertex is:
![x_{v} = -\frac{30}{2*3} = -5](https://tex.z-dn.net/?f=x_%7Bv%7D%20%3D%20-%5Cfrac%7B30%7D%7B2%2A3%7D%20%3D%20-5)
The y-value of the vertex is:
![f(-5) = 3(-5+5)^2 - 10 = -10](https://tex.z-dn.net/?f=f%28-5%29%20%3D%203%28-5%2B5%29%5E2%20-%2010%20%3D%20-10)
The vertex of f(x) is at (-5,-10).
Decreasing rate of g(x).
We have that:
1 - r = 0.95
So
r = 1 - 0.95 = 0.05
Which means that the decreasing rate is of 5%.
The zeros of h(x) are
The zeros are
x + 3 = 0 -> x = -3
x - 1 = 0 -> x = 1
So
The zeros of h(x) are at x = -3 and x = 1.