Question

I NEED HELP !!!!!!!!!!!!!!!!!!!!!! ( 20+ points )

Answer 1

1st one:

`\sf y - 0 = (1)/(3) (x - 1)` Or

`\sf y - (-1) = (1)/(3) (x - (-2))`

2nd one:

`\sf f(x) - 0 = (1)/(3) (x - 1)` Or

`\sf f(x) - (-1) = (1)/(3) (x - (-2))`

Explanation:

Let's take two points from the straight line.

(1,0) and (-2,-1)

Now,

To write the equation in point-slope form, we use the following formula:

`\sf y - y_1 = m(x - x_1)`

where
`\sf (x_1, y_1)` is one of the points on the line and m is the slope of the line.

In this case, we can use the point (1, 0) and the slope, which can be calculated as follows:

`\sf m = (y_2 - y_1)/(x_2 - x_1) \\\\ =(-1-0)/(-2-1)\\\\ =(1)/(3)`

Therefore, the equation in point-slope form is:

`\sf y - 0 = (1)/(3) (x - 1)`

If we use point (-2,-1), than the equation in point slope form will be:

`\sf y - (-1) = (1)/(3) (x - (-2))`

To write the equation in function notation, we simply replace y with f(x).

`\sf f(x) - 0 = (1)/(3) (x - 1)`

Or,

`\sf y - (-1) = (1)/(3) (x - (-2))`

Therefore, the function notation of the equation is:

`\sf f(x) - 0 = (1)/(3) (x - 1)`

Or

`\sf f(x) - (-1) = (1)/(3) (x - (-2))`

Answer 2

`\textsf{Point-slope form:}\quad y-0=(1)/(3)(x-1)`

`\textsf{Function notation:}\quad f(x)=(1)/(3)x-(1)/(3)`

Explanation:

The point-slope form of a linear equation is:

`\boxed{y-y_1=m(x-x_1)}`

where:

• (x₁, y₁) is a point on the line
• m is the slope of the line.

The slope of a line is a measure of how steep the line is and represents the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

The given graph shows two points on the line at (-2, -1) and (1, 0). Therefore, the ratio of the vertical change to the horizontal change between these two points is 1/3. So, the slope of the graphed line is m = 1/3.

To write the equation of the line in point-slope form, we can substitute the found slope m = 1/3 and one of the points (1, 0) into the point-slope formula.

`y-0=(1)/(3)(x-1)`

In function notation, we replace y with f(x) to represent that y is a function of x. Therefore, to rewrite the point-slope form in function notation, isolate y, then replace y with f(x).

Isolate y:

`y=(1)/(3)(x-1)`

`y=(1)/(3)x-(1)/(3)`

Replace y with f(x):

`f(x)=(1)/(3)x-(1)/(3)`

Therefore, the function notation of the linear equation is:

`f(x)=(1)/(3)x-(1)/(3)`