Question

family of lines
separate the graph paper into 4 sections and then graph each family on the same graph.
graph and explanations
observations
family
equation
slope
y - intercept
similarities
differences
1.
2.
3.
e
y = 1
e
y = -7
e
y = 5
e
y = -3
family
equation
slope
y - intercept
similarities
differences
1.
2.
3.
observation
f
x = 6
f
x = -4
f
x = 3
f
x = -1
family e
family f

Explanation:

Step1: Analyze Family E (Horizontal Lines)

Family E has equations \( y = 1 \), \( y = -7 \), \( y = 5 \), \( y = -3 \). For a horizontal line \( y = k \), the slope \( m = 0 \) (since there's no change in \( y \) for a change in \( x \)), and the y - intercept is \( (0,k) \).

• Slope: \( 0 \) for all equations.
• Y - intercept: For \( y = 1 \), y - intercept is \( 1 \); for \( y=-7 \), y - intercept is \( -7 \); for \( y = 5 \), y - intercept is \( 5 \); for \( y=-3 \), y - intercept is \( -3 \).
• Similarities: All lines are horizontal (slope \( 0 \)), parallel to each other.
• Differences: Different y - intercepts (different \( y \) - values).

Step2: Analyze Family F (Vertical Lines)

Family F has equations \( x = 6 \), \( x=-4 \), \( x = 3 \), \( x=-1 \). For a vertical line \( x = k \), the slope is undefined (since there's no change in \( x \) for a change in \( y \)), and there is no y - intercept (or it does not cross the y - axis in the traditional sense, except when \( x = 0 \), but these lines are not \( x = 0 \)).

• Slope: Undefined for all equations.
• Y - intercept: None (vertical lines do not have a y - intercept unless \( x = 0 \), which these are not).
• Similarities: All lines are vertical (undefined slope), parallel to each other.
• Differences: Different x - values (different positions on the x - axis).

Step3: Graphing

• For Family E (horizontal lines): Plot points for each \( y \) - value. For example, for \( y = 1 \), plot \( (0,1) \), \( (1,1) \), \( (2,1) \) etc. and draw a horizontal line. Do the same for \( y=-7 \), \( y = 5 \), \( y=-3 \).
• For Family F (vertical lines): Plot points for each \( x \) - value. For example, for \( x = 6 \), plot \( (6,0) \), \( (6,1) \), \( (6, - 1) \) etc. and draw a vertical line. Do the same for \( x=-4 \), \( x = 3 \), \( x=-1 \).

Step4: Observations

• For Family E: Horizontal lines are parallel, slope is zero, y - intercepts are the \( y \) - values of the equations.
• For Family F: Vertical lines are parallel, slope is undefined, no y - intercept (except \( x = 0 \) which is the y - axis itself), x - values are the positions of the vertical lines.
• Between Family E and F: Horizontal and vertical lines are perpendicular to each other (slope of horizontal is \( 0 \), slope of vertical is undefined, and their product is not defined in the usual slope - slope perpendicularity, but geometrically they are perpendicular).

Answer:

Family E (Horizontal Lines)

• Slope: \( 0 \) for all equations (\( y = 1,y=-7,y = 5,y=-3 \))
• Y - intercept: \( 1,-7,5,-3 \) respectively
• Similarities: All are horizontal (slope \( 0 \)), parallel
• Differences: Different y - intercepts

Family F (Vertical Lines)

• Slope: Undefined for all equations (\( x = 6,x=-4,x = 3,x=-1 \))
• Y - intercept: None
• Similarities: All are vertical (undefined slope), parallel
• Differences: Different x - values

Graphing

• Family E: Draw horizontal lines at \( y = 1,y=-7,y = 5,y=-3 \)
• Family F: Draw vertical lines at \( x = 6,x=-4,x = 3,x=-1 \)

Observations

• Family E: Horizontal, slope \( 0 \), y - intercept is the \( y \) - value of the equation.
• Family F: Vertical, slope undefined, no y - intercept, x - value is the position of the line.
• E and F lines: Perpendicular to each other.