Math Problem Statement
Graph the piecewise function: f(x) = (1/2)x^2 - 1 for x ≤ -2 and f(x) = -(1/2)x + 5 for x > -2.
Solution
The piecewise function provided is:
\begin{cases} \frac{1}{2}x^2 - 1 & \text{if } x \leq -2 \\ -\frac{1}{2}x + 5 & \text{if } x > -2 \end{cases}$$ To graph this function, follow these steps: ### 1. For $$x \leq -2$$, the function is quadratic: $$f(x) = \frac{1}{2}x^2 - 1$$ This is a parabola opening upwards because the coefficient of $$x^2$$ is positive ($$\frac{1}{2}$$). The vertex of the parabola occurs at $$x = 0$$, but we are only interested in the portion for $$x \leq -2$$. ### 2. For $$x > -2$$, the function is linear: $$f(x) = -\frac{1}{2}x + 5$$ This is a straight line with a slope of $$-\frac{1}{2}$$ and a y-intercept at 5. ### 3. Behavior at $$x = -2$$: At $$x = -2$$, check the function values from both pieces: - From the quadratic part: $$f(-2) = \frac{1}{2}(-2)^2 - 1 = 2 - 1 = 1$$. - From the linear part: $$f(-2)$$ is undefined in the linear part since it only applies for $$x > -2$$. Thus, the function is continuous at $$x = -2$$ with $$f(-2) = 1$$. ### Graph: - For $$x \leq -2$$, plot the parabola $$\frac{1}{2}x^2 - 1$$ starting at $$x = -2$$ (where $$f(-2) = 1$$). - For $$x > -2$$, plot the line $$-\frac{1}{2}x + 5$$. Let me know if you need a detailed graph, and I can generate it for you. Would you like further details or have any specific questions? --- ### Related Questions: 1. How do you determine the vertex of a quadratic function? 2. What does it mean for a function to be continuous at a point? 3. How do you find the y-intercept of a linear function? 4. Can you explain the process of graphing a piecewise function in more detail? 5. How can you determine if a piecewise function is differentiable at a point? ### Tip: When graphing piecewise functions, always check for continuity at the boundary points where the function changes its definition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Quadratic Functions
Linear Functions
Continuity
Formulas
Quadratic formula: ax^2 + bx + c
Linear formula: y = mx + b
Theorems
Continuity at x = -2
Suitable Grade Level
Grades 10-12
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