Question

1. $m(x) = \frac{-3}{x} - 4$

Explanation:

Step1: Identify the parent function

The function \( m(x)=\frac{-3}{x}-4 \) is a transformation of the parent rational function \( y = \frac{1}{x} \). The general form of a rational function transformation is \( y=\frac{a}{x - h}+k \), where \( a \) is the vertical stretch/compression and reflection, \( h \) is the horizontal shift, and \( k \) is the vertical shift. Here, \( a=-3 \), \( h = 0 \), and \( k=-4 \).

Step2: Analyze vertical and horizontal asymptotes

For the parent function \( y=\frac{1}{x} \), the vertical asymptote is \( x = 0 \) and the horizontal asymptote is \( y = 0 \). For \( m(x)=\frac{-3}{x}-4 \), the vertical asymptote remains \( x = 0 \) (since there is no horizontal shift, \( h = 0 \)) and the horizontal asymptote is \( y=k=-4 \) (vertical shift down by 4 units).

Step3: Analyze the direction and stretch

The coefficient \( a=-3 \) means the graph is reflected over the \( x \)-axis (because \( a<0 \)) and vertically stretched by a factor of 3 (since \( |a| = 3>1 \)).

Step4: Plot key points (optional for graphing)

We can choose some \( x \)-values (excluding \( x = 0 \)) to find corresponding \( y \)-values. For example:

• When \( x = 1 \), \( m(1)=\frac{-3}{1}-4=-3 - 4=-7 \)
• When \( x=-1 \), \( m(-1)=\frac{-3}{-1}-4 = 3-4=-1 \)
• When \( x = 3 \), \( m(3)=\frac{-3}{3}-4=-1 - 4=-5 \)
• When \( x=-3 \), \( m(-3)=\frac{-3}{-3}-4 = 1-4=-3 \)

These points help in sketching the graph. The graph should have two branches: one in the third quadrant (since for \( x>0 \), \( y=\frac{-3}{x}-4 \) is negative and for \( x<0 \), \( y=\frac{-3}{x}-4 \) is also negative, wait no: when \( x<0 \), \( \frac{-3}{x} \) is positive (negative divided by negative), then \( +(-4) \), so for \( x=-1 \), \( y=-1 \) (positive - 4), for \( x=-3 \), \( y=-3 \) (1 - 4). Wait, actually, when \( x>0 \), \( \frac{-3}{x} \) is negative, so \( y=\) negative - 4 (more negative). When \( x<0 \), \( \frac{-3}{x} \) is positive, so \( y=\) positive - 4 (can be positive or negative depending on the value). Wait, for \( x=-0.5 \), \( \frac{-3}{-0.5}=6 \), then \( 6-4 = 2 \), so \( ( - 0.5,2) \) is a point. So the left branch ( \( x<0 \)) is in the second quadrant (since \( x<0 \) and \( y = 2>0 \) for \( x=-0.5 \)) and the right branch ( \( x>0 \)) is in the fourth quadrant ( \( x>0 \), \( y=-7<0 \) for \( x = 1 \)).

Step5: Sketch the graph

Using the asymptotes \( x = 0 \) (vertical) and \( y=-4 \) (horizontal), the reflection, stretch, and key points, we can sketch the graph. The left branch ( \( x<0 \)) should approach \( x = 0 \) from the left (vertical asymptote) and \( y=-4 \) from above (since as \( x
ightarrow-\infty \), \( \frac{-3}{x}
ightarrow0 \), so \( y
ightarrow - 4 \) from above) and the right branch ( \( x>0 \)) should approach \( x = 0 \) from the right and \( y=-4 \) from below (since as \( x
ightarrow+\infty \), \( \frac{-3}{x}
ightarrow0 \), so \( y
ightarrow - 4 \) from below). The left branch is in the second quadrant (for small negative \( x \)) and the right branch is in the fourth quadrant (for small positive \( x \)).

(Note: Since the grid is provided, we can use the above analysis to plot the graph. The key features are vertical asymptote \( x = 0 \), horizontal asymptote \( y=-4 \), reflection over \( x \)-axis, vertical stretch by 3, and vertical shift down by 4.)

Answer:

To graph \( m(x)=\frac{-3}{x}-4 \):

1. Draw the vertical asymptote \( x = 0 \) (the \( y \)-axis) and the horizontal asymptote \( y=-4 \) (a horizontal line 4 units below the \( x \)-axis).
2. For the left branch ( \( x<0 \)): The graph is in the second quadrant (for small \( x<0 \)) and approaches \( x = 0 \) from the left and \( y=-4 \) from above. It is a reflection (over \( x \)-axis) and vertical stretch of the parent function's left branch.
3. For the right branch ( \( x>0 \)): The graph is in the fourth quadrant (for small \( x>0 \)) and approaches \( x = 0 \) from the right and \( y=-4 \) from below. It is a reflection (over \( x \)-axis) and vertical stretch of the parent function's right branch.

(If we were to describe the graph based on the grid, we can plot points like \( (1, - 7) \), \( (-1, - 1) \), \( (3, - 5) \), \( (-3, - 3) \), \( (-0.5,2) \) etc., and connect them following the asymptotes.)