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Use the table to answer the question.

x y
0 −5

1 −9

2 −13

3 −17

Which of the following equations correctly describes the function in the table using function notation?



(1 point)
Responses

f(x)=−5x−4

f of x is equal to negative 5 x minus 4

f(y)=−5x−4
f left parenthesis y right parenthesis equals negative 5 x minus 4

f(x)=−4x−5

1 answer

To determine which equation describes the function in the table correctly, we first need to identify the relationship between \( x \) and \( y \).

Let's look at the changes in \( y \) as \( x \) increases:

  • From \( x = 0 \) to \( x = 1 \), \( y \) changes from \(-5\) to \(-9\) (decreases by 4).
  • From \( x = 1 \) to \( x = 2 \), \( y \) changes from \(-9\) to \(-13\) (decreases by 4).
  • From \( x = 2 \) to \( x = 3 \), \( y \) changes from \(-13\) to \(-17\) (decreases by 4).

The consistent decrease of 4 in \( y \) suggests a linear relationship. The slope \( m \) can be calculated as \( -4 \) (since \( y \) decreases by 4 for every increase of 1 in \( x \)).

Now, we can use the point-slope form of the equation for a line. Starting from the point \( (0, -5) \):

\[ y - y_1 = m(x - x_1) \]

Using \( (0, -5) \):

\[ y + 5 = -4(x - 0) \] \[ y + 5 = -4x \] \[ y = -4x - 5 \]

If we want to express this in function notation, it would be:

\[ f(x) = -4x - 5 \]

Among the options provided:

  1. \( f(x) = -5x - 4 \) (this doesn't fit)
  2. \( f(y) = -5x - 4 \) (this doesn't fit)
  3. \( f(x) = -4x - 5 \) (this fits)

Therefore, the correct equation is:

f(x) = -4x - 5