Question

in exercises 1 and 2, graph the function. compare the graph with the graph of $f(x) = \frac{1}{x}$.

1. $g(x) = \frac{0.25}{x}$
2. $h(x) = \frac{-2}{x}$

Explanation:

For Problem 1: Graphing \( g(x)=\frac{0.25}{x} \) and Comparing with \( f(x)=\frac{1}{x} \)

Step 1: Recall the Parent Function

The parent function is \( f(x)=\frac{1}{x} \), which is a hyperbola with two branches: one in the first quadrant (where \( x>0,y>0 \)) and one in the third quadrant (where \( x<0,y<0 \)). The general form of a rational function of the form \( y = \frac{a}{x} \) is a vertical stretch/compression and reflection of \( f(x)=\frac{1}{x} \).

Step 2: Analyze \( g(x)=\frac{0.25}{x} \)

For \( g(x)=\frac{0.25}{x} \), we can write it as \( g(x)=0.25\times\frac{1}{x} \). Since \( 0.25=\frac{1}{4} \) (a positive constant less than 1), this represents a vertical compression of the graph of \( f(x)=\frac{1}{x} \) by a factor of \( \frac{1}{4} \).

To Graph \( g(x) \):

• Choose x-values: Pick values of \( x \) (positive and negative) to find corresponding \( y \)-values. For example:
• When \( x = 1 \), \( g(1)=\frac{0.25}{1}=0.25 \)
• When \( x = 4 \), \( g(4)=\frac{0.25}{4}=0.0625 \)
• When \( x=-1 \), \( g(-1)=\frac{0.25}{-1}=-0.25 \)
• When \( x = -4 \), \( g(-4)=\frac{0.25}{-4}=-0.0625 \)
• Plot the points and draw the two branches of the hyperbola. The branches will be closer to the x-axis and y-axis compared to \( f(x)=\frac{1}{x} \) because of the vertical compression.

For Problem 2: Graphing \( h(x)=\frac{-2}{x} \) and Comparing with \( f(x)=\frac{1}{x} \)

Step 1: Recall Transformations of \( f(x)=\frac{1}{x} \)

The general form \( y=\frac{a}{x} \): if \( a>0 \), the branches are in Q1 and Q3; if \( a<0 \), the branches are in Q2 and Q4 (reflection over the x-axis or y-axis). Also, the absolute value of \( a \) determines the vertical stretch/compression.

Step 2: Analyze \( h(x)=\frac{-2}{x} \)

For \( h(x)=\frac{-2}{x} \), we can write it as \( h(x)=-2\times\frac{1}{x} \).

• The negative sign indicates a reflection over the x-axis (or y-axis) of the parent function \( f(x)=\frac{1}{x} \).
• The factor \( 2 \) (since \( | - 2|=2 \)) indicates a vertical stretch by a factor of \( 2 \) (because \( 2>1 \)).

To Graph \( h(x) \):

• Choose x-values: Pick values of \( x \) (positive and negative) to find corresponding \( y \)-values. For example:
• When \( x = 1 \), \( h(1)=\frac{-2}{1}=-2 \)
• When \( x = 0.5 \), \( h(0.5)=\frac{-2}{0.5}=-4 \)
• When \( x=-1 \), \( h(-1)=\frac{-2}{-1}=2 \)
• When \( x=-0.5 \), \( h(-0.5)=\frac{-2}{-0.5}=4 \)
• Plot the points and draw the two branches of the hyperbola. The branches will be in the second quadrant ( \( x<0,y>0 \)) and fourth quadrant ( \( x>0,y<0 \)) (due to the negative sign) and will be "stretched" away from the axes compared to \( f(x)=\frac{1}{x} \) (due to the vertical stretch by factor 2).

Summary of Comparisons:

• For \( g(x)=\frac{0.25}{x} \): Vertical compression of \( f(x)=\frac{1}{x} \) by factor \( \frac{1}{4} \), same quadrants (Q1, Q3) as \( f(x) \), but closer to the axes.
• For \( h(x)=\frac{-2}{x} \): Vertical stretch by factor \( 2 \) and reflection over x-axis (so branches in Q2, Q4), farther from the axes than \( f(x) \) (because of stretch) and in opposite quadrants (due to reflection).

(Note: Since the problem asks to graph and compare, the above steps guide the graphing process. If we were to describe the final graph characteristics:

For \( g(x) \): Hyperbola with branches in Q1 and Q3, vertically compressed (narrower? No, actually, since the coefficient is less than 1, it's a compression, so the branches are closer to the axes. For example, at \( x = 1 \), \( f(1)=1 \) and \( g(1)=0.25 \), so the point (1,0.25) is closer to the x-axis than (1,1).

For \( h(x) \): Hyperbola with branches in Q2 and Q4 (because the coefficient is negative), and vertically stretched. At \( x = 1 \), \( f(1)=1 \) and \( h(1)=-2 \), so the point (1,-2) is farther from the x-axis (in the negative y-direction) than (1,1) is in the positive y-direction. At \( x=-1 \), \( f(-1)=-1 \) and \( h(-1)=2 \), so ( - 1,2) is farther from the x-axis (positive y-direction) than ( - 1,-1) is in the negative y-direction.)

Answer:

(for the comparison part, as the graphing is a visual task):

• For \( g(x)=\frac{0.25}{x} \): The graph of \( g(x) \) is a vertical compression of the graph of \( f(x)=\frac{1}{x} \) by a factor of \( \frac{1}{4} \). Both graphs have branches in the first and third quadrants, but \( g(x) \)'s branches are closer to the x - axis and y - axis.
• For \( h(x)=\frac{-2}{x} \): The graph of \( h(x) \) is a vertical stretch (by factor 2) and a reflection (over the x - axis or y - axis) of the graph of \( f(x)=\frac{1}{x} \). The branches of \( h(x) \) are in the second and fourth quadrants (unlike \( f(x) \)'s first and third quadrants), and they are farther from the x - axis and y - axis than \( f(x) \)'s branches.