Question

-4 -7.5
-1 0.75
2 9
5 17.25
-3 7
0 -2
4 -14
7 -23
enter your answer in each box.
line f: y = \boxed{} x + \boxed{}
line g: y = \boxed{}

Explanation:

To determine the equations of the lines \( f \) and \( g \), we use the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Line \( f \)

We use the points from the first table: \((-4, -7.5)\), \((-1, 0.75)\), \((2, 9)\), \((5, 17.25)\).

Step 1: Calculate the slope \( m \)

Using two points, e.g., \((-1, 0.75)\) and \((2, 9)\):

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 0.75}{2 - (-1)} = \frac{8.25}{3} = 2.75 = \frac{11}{4} $$

Step 2: Find the y-intercept \( b \)

Using the point \((2, 9)\) and \( m = \frac{11}{4} \):

$$ y = \frac{11}{4}x + b \implies 9 = \frac{11}{4}(2) + b \implies 9 = \frac{11}{2} + b \implies b = 9 - 5.5 = 3.5 = \frac{7}{2} $$

Thus, the equation for Line \( f \) is \( y = \frac{11}{4}x + \frac{7}{2} \) or \( y = 2.75x + 3.5 \).

Line \( g \)

We use the points from the second table: \((-3, 7)\), \((0, -2)\), \((4, -14)\), \((7, -23)\).

Step 1: Calculate the slope \( m \)

Using two points, e.g., \((0, -2)\) and \((4, -14)\):

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-14 - (-2)}{4 - 0} = \frac{-12}{4} = -3 $$

Step 2: Find the y-intercept \( b \)

Using the point \((0, -2)\) (which is the y-intercept):

$$ b = -2 $$

Thus, the equation for Line \( g \) is \( y = -3x - 2 \).

Final Answers

• Line \( f \): \( y = \boldsymbol{2.75}x + \boldsymbol{3.5} \) (or \( y = \frac{11}{4}x + \frac{7}{2} \))
• Line \( g \): \( y = \boldsymbol{-3}x + \boldsymbol{-2} \) (or \( y = -3x - 2 \))

Answer:

To determine the equations of the lines \( f \) and \( g \), we use the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Line \( f \)

We use the points from the first table: \((-4, -7.5)\), \((-1, 0.75)\), \((2, 9)\), \((5, 17.25)\).

Step 1: Calculate the slope \( m \)

Using two points, e.g., \((-1, 0.75)\) and \((2, 9)\):

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 0.75}{2 - (-1)} = \frac{8.25}{3} = 2.75 = \frac{11}{4} $$

Step 2: Find the y-intercept \( b \)

Using the point \((2, 9)\) and \( m = \frac{11}{4} \):

$$ y = \frac{11}{4}x + b \implies 9 = \frac{11}{4}(2) + b \implies 9 = \frac{11}{2} + b \implies b = 9 - 5.5 = 3.5 = \frac{7}{2} $$

Thus, the equation for Line \( f \) is \( y = \frac{11}{4}x + \frac{7}{2} \) or \( y = 2.75x + 3.5 \).

Line \( g \)

We use the points from the second table: \((-3, 7)\), \((0, -2)\), \((4, -14)\), \((7, -23)\).

Step 1: Calculate the slope \( m \)

Using two points, e.g., \((0, -2)\) and \((4, -14)\):

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-14 - (-2)}{4 - 0} = \frac{-12}{4} = -3 $$

Step 2: Find the y-intercept \( b \)

Using the point \((0, -2)\) (which is the y-intercept):

$$ b = -2 $$

Thus, the equation for Line \( g \) is \( y = -3x - 2 \).

Final Answers

• Line \( f \): \( y = \boldsymbol{2.75}x + \boldsymbol{3.5} \) (or \( y = \frac{11}{4}x + \frac{7}{2} \))
• Line \( g \): \( y = \boldsymbol{-3}x + \boldsymbol{-2} \) (or \( y = -3x - 2 \))