Question

problem 3. at a track meet, the distance (in meters) traveled by student a is represented by the equation y = 7x. the graph shows the distance traveled by student b. a. which student is faster? b. graph the equation that represents student a in the same coordinate plane as student b. compare and interpret the steepness of each graph.

Explanation:

Step1: Determine the slope - formula for speed

Speed is given by the slope of the distance - time graph. For a linear equation $y = mx + b$, the slope $m$ represents the rate of change. For Student A, the equation is $y = 7x$, so the slope (speed) $m_A=7$ m/s.

Step2: Calculate the slope for Student B from the graph

The graph of Student B is a straight - line. We can use two points on the line, say $(0,0)$ and $(7,0)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(7,0)$. The slope $m_B=\frac{0 - 0}{7 - 0}=0$ m/s. Since the line is going down, we take two non - zero points, say $(0,7)$ and $(7,0)$. Then $m_B=\frac{0 - 7}{7 - 0}=- 1$ m/s. The magnitude of the slope of Student B's graph is $|m_B| = 1$ m/s.

Step3: Compare the slopes

Since $|m_A|=7$ m/s and $|m_B| = 1$ m/s, and $7>1$, Student A is faster.

Step4: Write the equation for Student A in the general form for graphing comparison

The equation for Student A is $y = 7x$, which has a slope of 7 and a y - intercept of 0. To graph Student A on the same coordinate plane as Student B, we know that for every increase of 1 in the x - value (time), the y - value (distance) increases by 7.

Answer:

a. Student A is faster.
b. The equation for Student A is $y = 7x$. It has a steeper slope than the line representing Student B. The slope of Student A's line is 7, meaning the distance traveled increases by 7 meters for every 1 - second increase in time. The y - intercept is 0, indicating that Student A starts at the origin.