Question

the amount that tina can swim in kilometers, y, after training for x weeks can be modeled by the function y = 0.4x + 1.6. which sentence correctly interprets the y - intercept in the context of this problem? ○ tina can swim an additional 0.4 kilometers each week of training. ○ tina can swim an additional 4 kilometers each week of training. ○ the average distance that tina can swim is 1.6 kilometers. ○ tina could swim 1.6 kilometers before training.

Explanation:

Step1: Recall the slope - intercept form of a linear equation

The slope - intercept form of a linear equation is $y = mx + b$, where $m$ is the slope (rate of change) and $b$ is the $y$-intercept. In the given equation $y=0.4x + 1.6$, $x$ represents the number of weeks of training and $y$ represents the number of kilometers Tina can swim.

Step2: Interpret the $y$-intercept

The $y$-intercept occurs when $x = 0$. In the context of this problem, when $x = 0$, it means that Tina has not started training yet (0 weeks of training). Substituting $x = 0$ into the equation $y=0.4(0)+1.6$, we get $y = 1.6$. So, the $y$-intercept of 1.6 represents the number of kilometers Tina could swim before she started training (when the number of weeks of training $x = 0$).

Now let's analyze the other options:

• Option 1: The statement "Tina can swim an additional 0.4 kilometers each week of training" describes the slope ($m = 0.4$), not the $y$-intercept.
• Option 2: The slope is 0.4, not 4, so this option is incorrect.
• Option 3: The $y$-intercept is not an average distance, it's the distance when $x = 0$ (before training), so this option is incorrect.

Answer:

Tina could swim 1.6 kilometers before training.