could someone please help me on one of these and explian to me too? plz show work

Answer:

3) rate of change = -5/4

4) rate of change = 3

5) rate of change = 1

6) steepest slope = #4

7) Please see the attached graph and explanation below

8) Please see the attached graph and explanation below

Step-by-step explanation:

I will do questions 3, 4, 6, 7, and 8, and will let you work on question 5 (since it involves the same process of solving for the rate of change as questions 3 and 4).  However, the rate of change for question 5 is 1.  

The rate of change is essentially the same as the slope of a linear equation, where it represents the ratio of a change in y to a corresponding change in x.  

In order to solve for the rate of change, choose two points from the given graph.  

Let (x₁, y₁) = (0, 200)

     (x₂, y₂) = (80, 100)

Substitute these values into the following slope formula:

m = (y₂ - y₁)/(x₂ - x₁)

m = (100 - 200)/(80 - 0)

m = -100/80

Reduce to lowest terms by dividing the numerator and the denominator by 20/20:

m = -5/4  

Therefore, the rate of change is -5/4.  

Similar to what we did in question 3, choose two ordered pairs from the given table:

Let (x₁, y₁)  = (0, -1)

     (x₂, y₂) = (3, 8)  

Substitute these values into the following slope formula:

m = (y₂ - y₁)/(x₂ - x₁)

[tex]m = \frac{8 - (-1)}{3 - 0}[/tex]

[tex]m = \frac{8 + 1 }{3}[/tex]

[tex]m = \frac{9}{3}[/tex]

m = 3

Therefore, the rate of change from the given table is 3.

The positive-sloped lines whose rate of change has the highest value will have the steepest slope.  For lines with negative slopes (downward-tilting lines), the highest absolute value for its rate of change will have the steepest negative slope.  

Hence, the linear function with the steepest slope is #4 because the value of y changes by 3 units for every unit of change in x values.    

The linear equation, y = -6x - 4, where the slope, m = -6, and the y-intercept is (0, -4).  

Technically, two points are sufficient enough to connect and create a line with. Please see the attached screenshot of the graph for y = -6x - 4.

The given linear equation is in its standard form, Ax + By = C. It is easier to graph when the equation is in its slope-intercept form, y = mx + b.

In order to transform the given standard equation to slope-intercept form, start by subtracting 3x from both sides:  

3x - 4y = 12

3x -3x - 4y = -3x + 12

-4y = -3x + 12

Divide both sides by -4 to isolate y:

[tex]\frac{-4y}{-4} = \frac{-3x + 12}{-4}[/tex]

y = ¾x - 3  ⇒ This is the slope-intercept form where the slope, m = ¾, and the y-intercept is (0, -3).

Similar to how we graphed the equation from question 7, start by plotting the y-intercept (0, -3) on the graph. Then, use the slope, m = ¾ (3 units up, 4 units run), to plot other points on the graph.  Your next point occurs at the x-intercept, (4, 0). The x-intercept is the point on the graph where it crosses the x-axis.  Connect the two points to create a line that will represent the given equation, 3x - 4y = 12.

Please see the attached screenshot of the graph for 3x - 4y = 12.  

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