Question

q2
sketch the graph of a function with the following features:

1. the y - intercept is 7
2. the function is increasing for all values of x
3. the function is positive for x > - 3
4. as ( x

ightarrowinfty,f(x)
ightarrowinfty ) and as ( x
ightarrow-infty,f(x)
ightarrow-infty )

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.