Question

complete a table of values for the equation
x
y
domain
range
equation #8
slope
y - intercept

complete a table of values for the equation
x
y
domain
range
equation #11
slope
y - intercept

complete a table of values for the e

Explanation:

Step1: Analyze the Equation (Assume Equation #8 is \( y = 2x + 1 \))

For a linear equation \( y = mx + b \), \( m \) is slope, \( b \) is y - intercept. Domain (x - values) and range (y - values) for linear functions (non - vertical) are all real numbers (\( \mathbb{R} \)).
For \( y = 2x+1 \):

• Slope: In \( y = mx + b \), \( m = 2 \), so slope is \( 2 \).
• y - intercept: \( b = 1 \), so y - intercept is \( 1 \).
• Domain: All real numbers (\( (-\infty,\infty) \)) as there are no restrictions on \( x \) for a linear function.
• Range: All real numbers (\( (-\infty,\infty) \)) as for any real \( x \), \( y = 2x + 1 \) will give a real \( y \), and we can get any real \( y \) by solving \( x=\frac{y - 1}{2} \) for any real \( y \).
• Table of Values: Let's choose \( x=-2,-1,0,1,2 \)
• When \( x=-2 \), \( y=2(-2)+1=-4 + 1=-3 \)
• When \( x=-1 \), \( y=2(-1)+1=-2 + 1=-1 \)
• When \( x = 0 \), \( y=2(0)+1=1 \)
• When \( x = 1 \), \( y=2(1)+1=3 \)
• When \( x = 2 \), \( y=2(2)+1=5 \)

Step2: Analyze the Second Equation (Assume Equation #11 is \( y=\frac{3}{2}x-4 \))

For linear equation \( y = mx + b \):

• Slope: \( m=\frac{3}{2} \)
• y - intercept: \( b=-4 \)
• Domain: All real numbers (\( (-\infty,\infty) \)) (no restrictions on \( x \) for linear function)
• Range: All real numbers (\( (-\infty,\infty) \)) (we can solve \( x=\frac{2(y + 4)}{3} \) for any real \( y \))
• Table of Values: Let's choose \( x=-2,0,2,4,6 \)
• When \( x=-2 \), \( y=\frac{3}{2}(-2)-4=-3-4=-7 \)
• When \( x = 0 \), \( y=\frac{3}{2}(0)-4=-4 \)
• When \( x = 2 \), \( y=\frac{3}{2}(2)-4=3 - 4=-1 \)
• When \( x = 4 \), \( y=\frac{3}{2}(4)-4=6 - 4=2 \)
• When \( x = 6 \), \( y=\frac{3}{2}(6)-4=9 - 4=5 \)

Answer:

For Equation #8 (\( y = 2x + 1 \)):

• Slope: \( 2 \)
• y - intercept: \( 1 \)
• Domain: \( (-\infty,\infty) \)
• Range: \( (-\infty,\infty) \)
• Table (x,y): \((-2,-3),(-1,-1),(0,1),(1,3),(2,5)\)

For Equation #11 (\( y=\frac{3}{2}x-4 \)):

• Slope: \( \frac{3}{2} \)
• y - intercept: \( -4 \)
• Domain: \( (-\infty,\infty) \)
• Range: \( (-\infty,\infty) \)
• Table (x,y): \((-2,-7),(0,-4),(2,-1),(4,2),(6,5)\)