Question

1. the graph of a function is a line that passes through the points (0, 1) and (3, 10). write an equation in the form y = mx + b for this function. \boxed{y=3x+1} 6. a kite starts on the ground and slowly ascends into the sky. it flies at the same altitude for about 10 minutes and then quickly drops to the ground. sketch a graph of the behavior of the kite over time. graph of height vs time with a rising, flat, then falling line 7. raoul says that function b has a greater initial value. is raoul correct? justify your answer. function a graph of a line through (0,0) with slope ~0.67 function b \

$$\begin{tabular}{|c|c|c|c|c|c|} \\hline x & 0 & 2 & 4 & 6 & 8 \\\\ \\hline y & 3 & 6 & 9 & 12 & 15 \\\\ \\hline \\end{tabular}$$

\boxed{yes because a goes up by 2 and b goes up by 3} 8. a hot air balloon descends from an altitude of 2,000 feet at a constant rate of 90 feet per minute. the graph shows the altitude of the balloon over time. write a linear function in the form y = mx + b to represent the situation. graph of altitude vs time with a line from (0,2000) with slope -90 \boxed{90=4xt}

Explanation:

Problem 5

Step1: Find the slope \( m \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using points \((0,1)\) and \((3,10)\), we have \( x_1 = 0,y_1 = 1,x_2 = 3,y_2 = 10 \). So \( m=\frac{10 - 1}{3 - 0}=\frac{9}{3}=3 \).

Step2: Find the y - intercept \( b \)

The y - intercept \( b \) is the value of \( y \) when \( x = 0 \). From the point \((0,1)\), when \( x = 0 \), \( y = 1 \), so \( b = 1 \).

Step3: Write the equation

Using the slope - intercept form \( y=mx + b \), with \( m = 3 \) and \( b = 1 \), the equation is \( y = 3x+1 \).

The initial value of a function is the value of the function when \( x = 0 \) (the y - intercept). For Function A, when \( x = 0 \), \( y = 0 \) (from the graph). For Function B, when \( x = 0 \), \( y = 3 \) (from the table). Since \( 3>0 \), Function B has a greater initial value. So Raoul is correct.

Step1: Identify \( m \) and \( b \)

In the linear function \( y=mx + b \), \( b \) is the initial value (when \( x = 0 \)) and \( m \) is the rate of change. The balloon starts at an altitude of \( 2000 \) feet, so \( b = 2000 \). It descends at a rate of \( 90 \) feet per minute, so the slope \( m=- 90 \) (negative because the altitude is decreasing).

Step2: Write the equation

Using \( y=mx + b \) with \( m=-90 \) and \( b = 2000 \), the equation is \( y=-90x + 2000 \).

Answer:

\( y = 3x + 1 \)

Problem 7