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:
[tex]f(x) = ax^{2} + bx + c[/tex]
It's vertex is the point [tex](x_{v}, f(x_{v})[/tex]
In which
[tex]x_{v} = -\frac{b}{2a}[/tex]
If a<0, the vertex is a maximum point, that is, the maximum value happens at [tex]x_{v}[/tex], and it's value is [tex]f(x_{v})[/tex]
Decaying exponential function:
A decaying exponential function has the following format:
[tex]A(x) = A(0)(1-r)^{x}[/tex]
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 [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x - x_{1})*(x - x_{2})*...*(x-x_n)[/tex], in which a is the leading coefficient.
Vertex of f:
The function f is given by:
[tex]3(x+5)^2 - 10[/tex]
Expanding the calculations to place at the correct format:
[tex]3(x+5)^2 - 10 = 3(x^2 + 10x + 25) - 10 = 3x^2 + 30x + 75 - 10 = 3x^2 + 30x + 65[/tex]
Which means that
[tex]a = 3, b = 30, c = 65[/tex]
The x-value of the vertex is:
[tex]x_{v} = -\frac{30}{2*3} = -5[/tex]
The y-value of the vertex is:
[tex]f(-5) = 3(-5+5)^2 - 10 = -10[/tex]
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.
