Math Problem Statement
Graph this function: f(x) = -4 if -6 ≤ x < -2; f(x) = -(1/2)x + 3 if -2 ≤ x < 6.
Solution
The given function is a piecewise function defined as follows:
\begin{cases} -4 & \text{if } -6 \leq x < -2 \\ -\frac{1}{2}x + 3 & \text{if } -2 \leq x < 6 \end{cases}$$ ### Step-by-Step Breakdown: 1. **For $$-6 \leq x < -2$$**: The function is a constant: $$f(x) = -4$$. This means that for all values of $$x$$ in this interval, the function has a value of -4. So, you would draw a horizontal line at $$y = -4$$ for $$x$$ from -6 to -2 (excluding $$x = -2$$). 2. **For $$-2 \leq x < 6$$**: The function is linear: $$f(x) = -\frac{1}{2}x + 3$$. This is a line with a slope of -1/2 and a y-intercept of 3. For this interval, you plot the line starting from $$x = -2$$ (including this point) up to $$x = 6$$ (excluding $$x = 6$$). ### Key Points to Plot: - For $$x = -6$$ to $$x = -2$$, $$f(x) = -4$$. So, plot points like (-6, -4) and (-2, -4). - For $$x = -2$$, $$f(-2) = -\frac{1}{2}(-2) + 3 = 4$$. This gives the point (-2, 4). - For $$x = 6$$, $$f(6) = -\frac{1}{2}(6) + 3 = 0$$. The point is (6, 0), but it is not included in the graph (open circle). ### Plot: - The first segment is a horizontal line from $$(-6, -4)$$ to $$(-2, -4)$$. - The second segment is a straight line from $$(-2, 4)$$ to just before $$(6, 0)$$, with an open circle at $$(6, 0)$$. Would you like a graph plotted for this? Let me know if you'd like more details or explanations. --- Here are 5 related questions for further understanding: 1. How do you find the slope of a line? 2. What is the difference between open and closed intervals on a graph? 3. How do you determine whether a point is included or excluded in a graph of a function? 4. Can you graph other piecewise functions with different intervals and slopes? 5. How does the slope of a line affect its steepness on a graph? **Tip:** When graphing piecewise functions, always pay close attention to the interval boundaries to ensure you correctly include or exclude points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Graphing
Formulas
f(x) = -4 for -6 ≤ x < -2
f(x) = -(1/2)x + 3 for -2 ≤ x < 6
Theorems
Slope-Intercept Form of a Linear Equation
Suitable Grade Level
Grades 9-12