q2\nsketch the graph of a function with the following features:\n1) the y - intercept is 7\n2) the function…

q2\nsketch the graph of a function with the following features:\n1) the y - intercept is 7\n2) the function is increasing for all values of x\n3) the function is positive for x > - 3\n4) as ( x\rightarrowinfty,f(x)\rightarrowinfty ) and as ( x\rightarrow-infty,f(x)\rightarrow-infty )

q2\nsketch the graph of a function with the following features:\n1) the y - intercept is 7\n2) the function is increasing for all values of x\n3) the function is positive for x > - 3\n4) as ( x\rightarrowinfty,f(x)\rightarrowinfty ) and as ( x\rightarrow-infty,f(x)\rightarrow-infty )

Answer

Explanation:

Step1: Determine the type of function

Since the function is increasing for all (x), has end - behavior (x\to\infty,f(x)\to\infty) and (x\to-\infty,f(x)\to-\infty), a linear function (y = mx + b) (where (m>0)) is a good candidate.

Step2: Use the y - intercept

We know that the (y) - intercept (b = 7), so the function is of the form (y=mx + 7).

Step3: Use the zero - crossing information

The function is positive for (x>-3). Let (y = 0), then (0=mx+7), (x=-\frac{7}{m}). Since (x=-3) is the zero - crossing, we have (-\frac{7}{m}=-3), so (m=\frac{7}{3})

Step4: Sketch the graph

The function is (y=\frac{7}{3}x + 7).

  • Plot the (y) - intercept at the point ((0,7))
  • Use the slope (m = \frac{7}{3}) (from ((0,7)), move 3 units to the right and 7 units up to get another point ((3,14)))
  • Since the function is linear with positive slope, it is increasing for all (x). When (x=-3), (y=\frac{7}{3}\times(-3)+7=-7 + 7=0). For (x>-3), (y>0) and as (x\to\infty,y\to\infty) and as (x\to-\infty,y\to-\infty)

Answer:

Sketch the line (y=\frac{7}{3}x + 7) with (y) - intercept ((0,7)), passing through ((-3,0)) and having a positive slope.