Math Problem Statement

Carefully graph f(x) = sqrt(x) for 0 <= x < 4 and (x - 6)^2 - 2 for 4 <= x <= 10. Then, write a detailed slope statement that describes the changes in slope over the interval.

Solution

The graph above shows the piecewise function f(x)f(x) defined as:

\sqrt{x}, & 0 \leq x < 4 \\ (x - 6)^2 - 2, & 4 \leq x \leq 10 \end{cases}$$ ### Slope Analysis: 1. **For $$0 \leq x < 4$$** ($$f(x) = \sqrt{x}$$): - The slope, $$f'(x) = \frac{1}{2\sqrt{x}}$$, is positive and decreasing as $$x$$ increases. The slope starts off steep near $$x = 0$$ and gradually flattens as it approaches $$x = 4$$. This means the curve is concave down, slowing its rise. 2. **At $$x = 4$$**: - The function has a sharp transition at $$x = 4$$, marked by a discontinuity in the derivative (slope) because $$\sqrt{x}$$ and $$(x - 6)^2 - 2$$ meet with different slopes at this point. 3. **For $$4 \leq x \leq 10$$** ($$f(x) = (x - 6)^2 - 2$$): - The slope, $$f'(x) = 2(x - 6)$$, starts negative (since $$x - 6 < 0$$ for $$x < 6$$), indicating the graph is decreasing from $$x = 4$$ to $$x = 6$$. - After $$x = 6$$, the slope becomes positive, and the graph begins to increase. This indicates the parabola is concave up in this interval, with the minimum at $$x = 6$$. Do you have any questions or need further clarification? Here are five related questions for you to explore: 1. How would the graph change if the interval for $$\sqrt{x}$$ extended beyond $$x = 4$$? 2. Can you find the exact points where the slope of the function is zero? 3. What happens to the function if we shift the parabola horizontally by adding or subtracting a constant to $$x$$? 4. How does the second derivative help in understanding the concavity of each piece of the function? 5. How can we extend this piecewise function to make the transition at $$x = 4$$ smoother? **Tip**: When graphing piecewise functions, pay close attention to how the slopes (first derivatives) behave at the boundaries to identify any discontinuities in the slope.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Square Roots
Parabolas
Derivatives
Slope Analysis

Formulas

f(x) = sqrt(x)
f(x) = (x - 6)^2 - 2
Slope of sqrt(x): f'(x) = 1 / (2sqrt(x))
Slope of (x - 6)^2 - 2: f'(x) = 2(x - 6)

Theorems

Piecewise Function Definition
Derivative Definition

Suitable Grade Level

Grades 10-12