Math Problem Statement
Graph the piecewise function f(x) = { -1/4x - 3, for x < -4; -4, for -4 ≤ x ≤ 3; -x, for x > 3 }.
Solution
Let's analyze the piecewise function given in the image:
\begin{cases} -\frac{1}{4}x - 3, & \text{for } x < -4 \\ -4, & \text{for } -4 \leq x \leq 3 \\ -x, & \text{for } x > 3 \end{cases}$$ We will break down each part and match it with the appropriate graph: 1. **For $$x < -4$$**: The function is $$f(x) = -\frac{1}{4}x - 3$$. - This is a linear function with a slope of $$-\frac{1}{4}$$ and a y-intercept at $$-3$$. - The graph should show a line with a negative slope for values of $$x$$ less than $$-4$$. 2. **For $$-4 \leq x \leq 3$$**: The function is $$f(x) = -4$$. - This represents a horizontal line at $$y = -4$$ for the interval from $$x = -4$$ to $$x = 3$$. - The graph should show a flat line at $$y = -4$$ between these values. 3. **For $$x > 3$$**: The function is $$f(x) = -x$$. - This is a linear function with a slope of $$-1$$, passing through the origin if extended. - The graph should display a line with a slope of $$-1$$ for $$x > 3$$. Now, looking at the four graph options: - **Option A** shows a correct negative slope for $$x < -4$$, a horizontal line at $$y = -4$$, and a correct slope of $$-1$$ for $$x > 3$$. This option matches the piecewise function description. - **Option B** shows a different pattern for the slope, so it's incorrect. - **Option C** also differs in how it handles the lines, making it incorrect. - **Option D** is similarly inconsistent with the given piecewise function. Thus, **Option A** is the correct graph for this piecewise function. Would you like more detailed explanations for any of the steps? Here are some related questions to consider: 1. How do you find the slope and y-intercept of a linear equation? 2. What is the significance of the domain restrictions in a piecewise function? 3. How do you graph a constant function like $$f(x) = -4$$? 4. Can you explain how the slope of a line affects the steepness and direction? 5. What are some strategies for graphing piecewise functions efficiently? **Tip:** Always carefully check the boundaries (open or closed circles) in piecewise function graphs to ensure they match the function definition exactly.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Graphing
Formulas
f(x) = -1/4x - 3, for x < -4
f(x) = -4, for -4 ≤ x ≤ 3
f(x) = -x, for x > 3
Theorems
Slope-intercept form: y = mx + b
Domain restrictions for piecewise functions
Suitable Grade Level
Grades 9-12