If $y=x-6$ were changed to $y=x+8$, how would the graph of the new function compare with the first one?
A. It would be shifted right.
B. It would be shifted down.
C. It would be shifted up.
D. It would be steeper.
Answer (1)
The original function y = x − 6 has a y-intercept of -6.
The new function y = x + 8 has a y-intercept of 8.
The vertical shift is the difference in y-intercepts: 8 − ( − 6 ) = 14 .
The graph of the new function is shifted up by 14 units: C .
Explanation
Analyze the functions The original function is y = x − 6 . The new function is y = x + 8 . Both functions are linear functions with a slope of 1. The y-intercept of the original function is -6. The y-intercept of the new function is 8.
Compare the y-intercepts The graphs of these functions are straight lines. The slope of both lines is 1, which means they have the same steepness. The only difference between the two functions is the y-intercept. The y-intercept of y = x − 6 is -6, and the y-intercept of y = x + 8 is 8.
Calculate the vertical shift To find the vertical shift, we subtract the y-intercept of the original function from the y-intercept of the new function:
Calculate the difference 8 − ( − 6 ) = 8 + 6 = 14
Determine the direction of the shift Since the difference is positive (14), the new graph is shifted up by 14 units relative to the original graph.
Conclusion Therefore, the graph of the new function would be shifted up compared to the first one.
Examples
Imagine you're adjusting the thermostat in your house. The original function y = x − 6 represents the temperature setting you initially had, where 'x' is the desired temperature and 'y' is the actual temperature achieved (with a slight offset). If you change the setting to y = x + 8 , you're essentially shifting the entire temperature range upwards. This is similar to adjusting all the temperatures in your house by a fixed amount, making everything warmer by a consistent value. Understanding how these shifts work helps you fine-tune your environment to your liking.
