Question

How to complete Number 2 of this Question? Algebra 2

Answer 1

##2)

The equation given is

`y=-2x+4`Domain

The domain is the set of x-values for which the function is defined.

A linear function is defined for all values of x. Thus, the domain is

`D=(-\infty,\infty)`Range

The range is the set of y-values for which the function is defined.

A linear function is defined for all values of y. Thus, the range is

`R=(-\infty,\infty)`Zeros and x-intercept

The zero(s) and the x-intercept are essentially the same thing. This is the point where the line cuts the x-axis. We find this by setting y = 0. Shown below:

`\begin{gathered} y=-2x+4 \\ 0=-2x+4 \\ 2x=4 \\ x=(4)/(2) \\ x=2 \end{gathered}`

The zero is x = 2

The x-intercept is (2, 0)

y-intercept

The y-intercept of a line is the point where the line cuts the y-axis. It is found by setting x = 0. Shown below:

`\begin{gathered} y=-2x+4 \\ y=-2(0)+4 \\ y=0+4 \\ y=4 \end{gathered}`

The y-intercept is y = 4.

Continuity

A linear function is continuous at all points. So, the line is continuous at the interval:

`(-\infty,\infty)`

Increase Interval

This is the interval on which the graph of the line is increasing.

The graph of the line shows that the line is always decreasing. So, it doesn't increase on any interval.

`\emptyset`

Decrease Interval

This is the interval on which the graph of the line is decreasing.

The graph of this line shows that the line is always decreasing.

So, the interval on which the line is decreasing is:

`(-\infty,\infty)`Maximum and Minimum

A straight line cannot have maximum or minimum unless we are given an interval.

Since we aren't given any interval. There is no min or max of a line.

Value of f(-1) and f(15)

We have to find the values of the function (y = -2x + 4) at x = -1 and x = 15.

Let's write the equation of the line in functional notation:

`\begin{gathered} y=-2x+4 \\ f(x)=-2x+4 \end{gathered}`

We will substitute x = - 1 and x = 15 into the function and find the corresponding values. Shown below:

`\begin{gathered} f(x)=-2x+4 \\ --------------- \\ f(-1)=-2(-1)+4 \\ f(-1)=6 \\ --------------- \\ f(15)=-2(15)+4 \\ f(15)=-26 \end{gathered}`

As x → - ∞ and x → ∞

We have to find "where" the function (line) approaches as x approaches negative infinity and positive infinity.

From the graph, we clearly see that as x → - ∞, y → ∞, and

as x → ∞, y → - ∞

So,

`\begin{gathered} x\rightarrow-\infty,y\rightarrow\infty \\ \text{and} \\ x\rightarrow\infty,y\rightarrow-\infty \end{gathered}`Vertical Asymptote and Horizontal Asymotote

Vertical asymptotes occur where the denominator becomes zero as long as there are no common factors.

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.

A line doesn't have any vertical or horizontal asymptotes.