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 need to determine which of the given equations has the least steep graph. The equations are in the slope-intercept form, y = m x + b , where m represents the slope and b is the y-intercept. The steepness of the graph is determined by the absolute value of the slope ∣ m ∣ . The smaller the absolute value of the slope, the less steep the 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, let's 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 Steepness Comparing the absolute values of the slopes, 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 , which corresponds to equation D.
Final Answer Therefore, the equation with the least steep graph is D. y = 4 1 x + 9 .
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 vehicles can safely travel. A smaller slope (less steep) is easier to navigate, while a larger slope (more steep) might require more power or effort. Similarly, in finance, the slope of a trend line can indicate the rate of growth or decline of an investment. By comparing the slopes of different investment options, investors can make informed decisions about where to allocate their resources. In general, the concept of slope helps us quantify and compare rates of change in various scenarios.
