Math Problem Statement
Which of the following is the graph of the function f(x) = {(x + 3)^2 - 4 ; x ≤ -3, x + 1 ; -3 < x ≤ 1, (x - 2)^2 - 4 ; x > 1}?
Solution
We are given a piecewise function:
\begin{cases} (x + 3)^2 - 4 & \text{if } x \leq -3 \\ x + 1 & \text{if } -3 < x \leq 1 \\ (x - 2)^2 - 4 & \text{if } x > 1 \end{cases}$$ To identify which graph corresponds to this function, let's break down each piece and its behavior: 1. **First piece: $$f(x) = (x + 3)^2 - 4$$, for $$x \leq -3$$:** - This is a parabola shifted left by 3 and down by 4. - It opens upwards because of the positive coefficient of $$x^2$$. - For $$x = -3$$, the vertex is at $$(-3, -4)$$. 2. **Second piece: $$f(x) = x + 1$$, for $$-3 < x \leq 1$$:** - This is a straight line with slope 1 and y-intercept at $$1$$. - The line starts from $$(-3, -2)$$ (since at $$x = -3$$, $$f(-3) = -3 + 1 = -2$$) and goes up to $$(1, 2)$$. 3. **Third piece: $$f(x) = (x - 2)^2 - 4$$, for $$x > 1$$:** - This is a parabola shifted right by 2 and down by 4. - It opens upwards as well. - For $$x = 2$$, the vertex is at $$(2, -4)$$. ### Now, analyzing the graphs: - **Graph A**: This graph has discontinuities and the second piece doesn’t match the line segment given by $$f(x) = x + 1$$. So, this is not the correct graph. - **Graph B**: This graph appears to show a smooth transition from a parabola to a linear function between $$x = -3$$ and $$x = 1$$, and then to another parabola for $$x > 1$$. The features match the breakdown of the function. - **Graph C**: The linear part doesn’t connect properly at $$x = -3$$, and the shape of the parabolas doesn't seem correct. - **Graph D**: The linear piece does not align correctly with the endpoints and the parabola for $$x \leq -3$$ is incorrect. ### Conclusion: The correct graph is **B**. Would you like further details on how each function transforms the graph? Here are 5 related questions to deepen your understanding: 1. What is the domain and range of the piecewise function? 2. How do you determine if a piecewise function is continuous? 3. Can piecewise functions have different types of functions in each interval, such as a line and a parabola? 4. How do shifts and reflections affect the graphs of functions like parabolas? 5. How can you check if the points where the pieces of the function meet are connected? **Tip**: When graphing piecewise functions, always pay attention to whether the pieces are defined with strict inequalities (open circles) or inclusive inequalities (closed circles) to see if the endpoints connect.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Parabolas
Linear Functions
Formulas
(x + 3)^2 - 4 for x ≤ -3
x + 1 for -3 < x ≤ 1
(x - 2)^2 - 4 for x > 1
Theorems
Continuity of Piecewise Functions
Parabola Shifts
Linear Functions
Suitable Grade Level
Grades 9-12