Mathematics: (a) find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) find an equation for the family of linear functions such that (2) = 1 and sketch several members ofthe family. (c) which function belongs to both families?

x divided by 75% = yStep-by-step explanation:...

(a) find an equation for the family of linear

Ответ:

gabypinskyb8045

15.01.2022

Mathematics

$\textbf{(\text{a})}$ f(x) = 2x+n

$\textbf{(\text{b})}\\$ f(x) = mx + 1 -2m = m(x-2) + 1

$\textbf{(\text{c})}$ f(x) = 2x - 3

Step-by-step explanation:

First, we know that family of functions represents a set of functions whose equations have a similar form. In our case, a family of linear functions can be represented as

$\{ ax + b | a, b \in \mathbb{R} \}$ .

Now, we can take an arbitrary member of that family, a function

f(x) = mx + n

for some real constants and .

$\textbf{(\text{a})}$

In this part of the problem, we know that $m = 2\\$ , so we consider

f(x) = 2x +n .

To graph several members of the family, you can plug in any real number in the equation above instead of , since $\forall n \in \mathbb{R}$ satisfy the equation.

For n = 0 , we have f(x) = 2x.

For n = 15 , we have f(x) = 2x + 15 .

For n = -15 , we have f(x) = 2x - 15.

The graphs for the values n = 0, n = 15 and n = -15 are presented on the first graph below.

$\textbf{(\text{b})}$

We need to find the member of the family of linear functions such that

f(2) = 1 .

Substituting for in f(x) = mx + n gives

f(2) = 2m+n .

Now, since we have that f(2) = 1 , we can equate with 2m +n and express one of them in terms of the other.

$2m + n = 1 \implies n = 1 - 2m$

Substituting 1-2m for in f(x) = mx + n gives the equation

f(x) = mx + 1 -2m = m(x-2) + 1

which represents the wanted family. To sketch several member, we can choose any real value for , since $\forall m \in \mathbb{R}$ satisfy the equation.