- Identify the slope \(m\) and y-intercept \(b\) for both functions: \(f(x) = \frac{1}{4}x - 1\) and \(g(x) = \frac{1}{2}x - 2\).
- Compare the absolute values of the slopes: \(|0.5| > |0.25|\), confirming that \(g(x)\) is steeper.
- Compare the y-intercepts: \(-2 < -1\), confirming that the y-intercept of \(g(x)\) is lower.
- The graph of \(g(x)\) is steeper and has a lower \(y\)-intercept. \(\boxed{\text{The line of } g(x) \text{ is steeper and has a lower } y\text{-intercept.}}\)
Explanation
- Analyzing the linear functions
The function \(f(x) = \frac{1}{4}x - 1\) represents a line in the coordinate plane. Comparing it to the standard form of a linear equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, we can see that for \(f(x)\), the slope \(m_f = \frac{1}{4}\) and the y-intercept \(b_f = -1\). The new function \(g(x) = \frac{1}{2}x - 2\) also follows this form, with a slope \(m_g = \frac{1}{2}\) and a y-intercept \(b_g = -2\).
- Comparing steepness
The steepness of a line is determined by the absolute value of its slope \(|m|\). Comparing the slopes of the two functions:
- For \(f(x)\), \(|m_f| = |\frac{1}{4}| = 0.25\).
- For \(g(x)\), \(|m_g| = |\frac{1}{2}| = 0.5\).
Since \(0.5 > 0.25\), the absolute value of the slope for \(g(x)\) is greater than that for \(f(x)\), which means the graph of \(g(x)\) is steeper than the graph of \(f(x)\).
- Comparing y-intercepts
The y-intercept represents the point where the line crosses the y-axis, given by the value of \(b\) when \(x = 0\).
- For \(f(x)\), the y-intercept is \(-1\).
- For \(g(x)\), the y-intercept is \(-2\).
Since \(-2 < -1\), the y-intercept of \(g(x)\) is lower than the y-intercept of \(f(x)\).
- Conclusion
Based on the analysis of the slopes and y-intercepts, the graph of \(g(x)\) is steeper than the graph of \(f(x)\) and has a lower y-intercept. Thus, the correct description is that the line of \(g(x)\) is steeper and has a lower \(y\)-intercept.
Examples
Consider road construction where you are designing an incline or a ramp. The steepness of the ramp is modeled by the slope of a line. If you change the design of a ramp from a slope of \(\frac{1}{4}\) (for every 4 meters of horizontal distance, the ramp rises 1 meter) to a slope of \(\frac{1}{2}\) (for every 2 meters of horizontal distance, the ramp rises 1 meter), the ramp becomes significantly steeper and harder to climb. Similarly, the y-intercept represents the starting height of the ramp. A lower y-intercept means the ramp starts from a deeper point in the ground or a lower platform. Understanding these changes helps engineers and designers accurately modify structures for accessibility or safety.