Question

Solve Linear Functions Represented in Different Ways and Write about the Solutions.
Compare the linear functions expressed by the equation, y = –x + 3, and by data in the table.

A 2-column table with 4 rows. Column 1 is labeled x with entries negative 4, negative 2, 1, 3. Column 2 is labeled y with entries 1, negative 1, negative 4, negative 6.

Explain how to determine if these two are the same function expressed in different ways

Answer 1

To check whether the table represents the same line as y = −x + 3, you can either (A) test whether each table point satisfies y = −x + 3, or (B) find the line determined by the table and compare slopes and intercepts.

A. Test by substitution
- For x = −4, y_pred = −(−4) + 3 = 4 + 3 = 7, but table gives y = 1 → does not match.
Because one table point fails, the table is not the same as y = −x + 3.

B. Find the line from the table
- Compute slopes between consecutive points:
- Between (−4,1) and (−2,−1): m = (−1 − 1)/(−2 − (−4)) = −2/2 = −1
- Between (−2,−1) and (1,−4): m = (−4 − (−1))/(1 − (−2)) = −3/3 = −1
- Between (1,−4) and (3,−6): m = (−6 − (−4))/(3 − 1) = −2/2 = −1
Slope is constant m = −1, so the table points lie on a line with slope −1.
- Find the y-intercept b using one point, e.g. (−4,1): 1 = (−1)(−4) + b → 1 = 4 + b → b = −3.
So the line from the table is y = −x − 3.

Compare the two lines:
- Given: y = −x + 3 (slope −1, intercept +3)
- From table: y = −x − 3 (slope −1, intercept −3)

They have the same slope but different y-intercepts, so they are parallel lines, not the same function.