Which equation has the steepest graph?
A. [tex]$y=\frac{3}{4} x-9$[/tex]
B. [tex]$y=-14 x+1$[/tex]
C. [tex]$y=2 x+8$[/tex]
D. [tex]$y=10 x-5$[/tex]
Answer (1)
Find the slope of each equation.
Take the absolute value of each slope.
Compare the absolute values of the slopes.
The equation with the largest absolute value of the slope has the steepest graph, which is B .
Explanation
Understanding the Problem We are given four linear equations in slope-intercept form: y = m x + b , where m is the slope and b is the y-intercept. The steepness of the graph is determined by the absolute value of the slope.
Finding the Slope of Equation A The slope of equation A is 4 3 . The absolute value of the slope is 4 3 = 4 3 = 0.75 .
Finding the Slope of Equation B The slope of equation B is − 14 . The absolute value of the slope is ∣ − 14∣ = 14 .
Finding the Slope of Equation C The slope of equation C is 2 . The absolute value of the slope is ∣2∣ = 2 .
Finding the Slope of Equation D The slope of equation D is 10 . The absolute value of the slope is ∣10∣ = 10 .
Comparing the Slopes and Finding the Steepest Graph Comparing the absolute values of the slopes, we have 0.75 < 2 < 10 < 14 . Therefore, equation B has the steepest graph.
Final Answer The equation with the steepest graph is B. y = − 14 x + 1 .
Examples
Understanding the steepness of a line is crucial in many real-world applications. For instance, when designing roads or ramps, engineers need to consider the slope to ensure safety and ease of use. A steeper slope means a more challenging climb, which can affect the type of vehicles that can use the road or the effort required to ascend a ramp. In finance, the slope of a stock's price trend indicates how quickly its value is changing, helping investors make informed decisions. Similarly, in physics, the slope of a velocity-time graph represents acceleration, a key concept in understanding motion.
