Question

a consumer group is investigating the relationship between sugar content and number of calories per serving. the equation of the line of best fit for the data is ( y = 4x + 80 ), where ( x ) is the number of grams of sugar. what is the meaning of the ( y )-intercept in the line of best fit?

a. out of all the brands of cereal, the greatest number of calories in one serving was 80.

b. the number of calories in one serving is estimated to be 80 if there is 0 grams of sugar.

c. out of all the brands of cereal, there were 0 brands of cereal that had 80 calories in one serving.

d. the number of calories in one serving is expected to increase by 80 for each increase in 1 gram of sugar.

Explanation:

Step1: Recall slope - intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. In the given equation \(y=4x + 80\), \(x\) represents the number of grams of sugar and \(y\) represents the number of calories.

Step2: Interpret the \(y\) - intercept

The \(y\) - intercept (\(b = 80\)) occurs when \(x = 0\) (since at the \(y\) - intercept, the \(x\) - value is \(0\)). Substituting \(x = 0\) into the equation \(y=4x + 80\), we get \(y=4(0)+80=80\). This means that when the number of grams of sugar (\(x\)) is \(0\), the estimated number of calories (\(y\)) in one serving is \(80\).

Let's analyze each option:

• Option A: The equation is about the relationship between sugar and calories, not about the number of brands and calories. So A is incorrect.
• Option B: When \(x = 0\) (0 grams of sugar), \(y = 80\), which means the number of calories in one serving is estimated to be 80 when there is 0 grams of sugar. This matches the interpretation of the \(y\) - intercept.
• Option C: The \(y\) - intercept has nothing to do with the number of brands of cereal. So C is incorrect.
• Option D: The slope (\(m = 4\)) represents the rate of change of \(y\) with respect to \(x\), i.e., the number of calories increases by 4 for each 1 - gram increase in sugar. The \(y\) - intercept is not related to the slope's interpretation. So D is incorrect.

Answer:

B. The number of calories in one serving is estimated to be 80 if there is 0 grams of sugar.