If $y=12x-7$ were changed to $y=12x+1$, how would the graph of the new function compare with the original?
A. It would be shifted down.
B. It would be less steep.
C. It would be shifted up.
D. It would be steeper.
Answer (1)
The original function is y = 12 x − 7 and the new function is y = 12 x + 1 .
Both functions have the same slope, which is 12.
The y-intercept of the original function is -7, and the y-intercept of the new function is 1.
Since the y-intercept of the new function is greater than the y-intercept of the original function, the graph of the new function is shifted up. The answer is C .
Explanation
Understanding the Functions We are given two linear functions: the original function y = 12 x − 7 and the new function y = 12 x + 1 . We need to determine how the graph of the new function compares to the graph of the original function.
Identifying Slope and Y-intercept Both functions are in the slope-intercept form, y = m x + b , where m represents the slope and b represents the y-intercept. For the original function, the slope is 12 and the y-intercept is -7. For the new function, the slope is also 12, and the y-intercept is 1.
Comparing Slopes and Y-intercepts Since the slopes of both functions are the same (12), the lines are parallel, meaning they have the same steepness. The only difference between the two functions is their y-intercepts. The original function has a y-intercept of -7, while the new function has a y-intercept of 1.
Determining the Shift The y-intercept of the new function (1) is greater than the y-intercept of the original function (-7). This means that the graph of the new function is shifted upwards compared to the graph of the original function. The vertical shift is 1 − ( − 7 ) = 8 units.
Conclusion Therefore, the graph of the new function would be shifted up compared to the original function. The correct answer is C.
Examples
Understanding how changing the y-intercept of a linear function affects its graph is useful in many real-world scenarios. For example, if a company's profit is modeled by a linear function, increasing the constant term (y-intercept) represents a direct increase in profit, shifting the entire profit graph upwards. Similarly, in physics, if you're analyzing the motion of an object with a constant velocity, changing the initial position (y-intercept) would shift the entire position-time graph up or down, reflecting the object's starting point.
