Use the table to answer the question. x y 0 3 1 31/2 2 4 3 41/2 Which of the following equations describes the function in the table using function notation? (1 point) Responses f(y)=3x+12 f left parenthesis y right parenthesis equals 3 x plus Start Fraction 1 over 2 End Fraction f(x)=3x+12 f left parenthesis x right parenthesis equals 3 x plus Start Fraction 1 over 2 End Fraction f(y)=12x+3 f left parenthesis y right parenthesis equals Start Fraction 1 over 2 End Fraction x plus 3 f(x)=12x+3

1 answer

To determine which equation describes the function in the table, we need to analyze the relationship between x and y in the provided data points. The given points are:

  • (0, 3)
  • (1, 3.5)
  • (2, 4)
  • (3, 4.5)

From these points, we can observe how y changes as x changes. Let's list the changes in y:

  • From x = 0 to x = 1, y changes from 3 to 3.5 (an increase of 0.5).
  • From x = 1 to x = 2, y changes from 3.5 to 4 (an increase of 0.5).
  • From x = 2 to x = 3, y changes from 4 to 4.5 (an increase of 0.5).

This suggests a linear relationship, as y is increasing by a constant amount (0.5) for each unit increase in x.

To find the equation, let's consider the slope. The slope (m) can be calculated as:

m = (change in y) / (change in x) = 0.5 / 1 = 0.5.

Now, we can use the point-slope form of a linear equation. We know one point is (0, 3). Using the slope-intercept form \(y = mx + b\), we substitute m and b (where b is the y-intercept) to get:

\(y = 0.5x + 3.\)

In function notation, this would be represented as:

\(f(x) = \frac{1}{2}x + 3.\)

Now let's check the answer choices:

  • f(y) = 12x + 3 (not fitting).
  • f(y) = \frac{1}{2}x + 3 (this fits).
  • f(x) = 12x + 3 (not fitting).
  • f(x) = 3x + \frac{1}{2} (not fitting).

The equation that describes the function in the table using function notation is:

f(y) = \frac{1}{2}x + 3.