Question

Identify the key features of these functions:

Answer 1

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:

$f(x) = ax^{2} + bx + c$

It's vertex is the point $(x_{v}, f(x_{v})$

In which

$x_{v} = -\frac{b}{2a}$

If a<0, the vertex is a maximum point, that is, the maximum value happens at $x_{v}$, and it's value is $f(x_{v})$

Decaying exponential function:

A decaying exponential function has the following format:

$A(x) = A(0)(1-r)^{x}$

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 $x_{1}, x_{2}, x_{n}$ such that it can be written as: $a(x - x_{1})*(x - x_{2})*...*(x-x_n)$, in which a is the leading coefficient.

Vertex of f:

The function f is given by:

$3(x+5)^2 - 10$

Expanding the calculations to place at the correct format:

$3(x+5)^2 - 10 = 3(x^2 + 10x + 25) - 10 = 3x^2 + 30x + 75 - 10 = 3x^2 + 30x + 65$

Which means that

$a = 3, b = 30, c = 65$

The x-value of the vertex is:

$x_{v} = -\frac{30}{2*3} = -5$

The y-value of the vertex is:

$f(-5) = 3(-5+5)^2 - 10 = -10$

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.