If $y=2 x+1$ were changed to $y=\frac{1}{2} x+1$, how would the graph of the new function compare with the first one?
A. It would be shifted left.
B. It would be less steep.
C. It would be shifted down.
D. It would be steeper.
Answer (1)
The original function is y = 2 x + 1 , and the new function is y = 2 1 x + 1 .
Both functions are in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept.
The slope of the original function is 2, and the slope of the new function is 2 1 .
Since 2 1 < 2 , the new line is less steep than the original line. The answer is B: It would be less steep.
Explanation
Understanding the Equations We are given two linear equations: y = 2 x + 1 and y = 2 1 x + 1 . We need to determine how the graph of the second equation compares to the first.
Slope-Intercept Form Both equations are in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y-intercept.
Identifying Slopes and Y-Intercepts For the first equation, y = 2 x + 1 , the slope m 1 = 2 and the y-intercept b 1 = 1 . For the second equation, y = 2 1 x + 1 , the slope m 2 = 2 1 and the y-intercept b 2 = 1 .
Comparing Slopes and Y-Intercepts Comparing the slopes, we see that m 1 = 2 and m 2 = 2 1 . Since 2 1 < 2 , the second line is less steep than the first line. Both lines have the same y-intercept, which is 1, so there is no vertical shift. The change in slope does not cause a horizontal shift.
Conclusion Therefore, the graph of the new function would be less steep compared to the first one.
Examples
Understanding the slope of a line is crucial in many real-world applications. For instance, when designing a ramp, the slope determines how easy it is to ascend. A smaller slope (less steep) makes the ramp easier to climb, while a larger slope (steeper) makes it more challenging. Similarly, in economics, the slope of a supply or demand curve indicates how responsive the quantity supplied or demanded is to changes in price. A flatter (less steep) curve suggests a smaller change in quantity for a given price change, while a steeper curve indicates a larger change.
