Which equation has the least steep graph?
A. $y=-\frac{1}{2} x-3$
B. $y=-6 x+1$
C. $y=2 x-7$
D. $y=\frac{1}{4} x+9$
Answer (1)
Identify the slopes of the given equations.
Calculate the absolute value of each slope.
Compare the absolute values of the slopes.
The equation with the smallest absolute value of the slope has the least steep graph, which is D .
Explanation
Understanding the Problem We are given four linear equations in slope-intercept form ( y = m x + b ), where m represents the slope and b represents the y-intercept. The steepness of the graph is determined by the absolute value of the slope, ∣ m ∣ . Our goal is to identify the equation with the smallest ∣ m ∣ , as this corresponds to the least steep graph.
Identifying Slopes Let's identify the slopes of the given equations:
A. y = − 2 1 x − 3 has a slope of m A = − 2 1 .
B. y = − 6 x + 1 has a slope of m B = − 6 .
C. y = 2 x − 7 has a slope of m C = 2 .
D. y = 4 1 x + 9 has a slope of m D = 4 1 .
Calculating Absolute Values of Slopes Now, we calculate the absolute values of the slopes:
A. ∣ m A ∣ = ∣ − 2 1 ∣ = 2 1 = 0.5 B. ∣ m B ∣ = ∣ − 6∣ = 6 C. ∣ m C ∣ = ∣2∣ = 2 D. $|m_D| = |\frac{1}{4}| = \frac{1}{4} = 0.25
Comparing and Concluding Comparing the absolute values, we have: ∣ m A ∣ = 0.5 ∣ m B ∣ = 6 ∣ m C ∣ = 2 ∣ m D ∣ = 0.25
The smallest absolute value is ∣ m D ∣ = 0.25 . Therefore, equation D has the least steep graph.
Examples
Understanding the steepness of a line is crucial in many real-world applications. For example, when designing roads or ramps, engineers need to consider the slope to ensure safety and ease of use. A smaller slope means a less steep incline, which is easier for vehicles and pedestrians to navigate. In finance, the slope of a trend line can indicate the rate of growth or decline of an investment. Therefore, understanding the concept of slope and its absolute value is essential in various fields.
