Math Problem Statement
Graph the following piecewise function on the axes provided:
\[
f(x) = \begin{cases}
2x + 5 & \text{for } x \leq -1 \\
-2 & \text{for } x > 5
\end{cases}
\]
Solution
Let's analyze the given function and determine how to graph it step by step. The function is defined as:
\begin{cases} 2x + 5 & \text{for } x \leq -1 \\ -2 & \text{for } x > 5 \end{cases}$$ --- ### **Step 1: Analyze the first case $$f(x) = 2x + 5$$ for $$x \leq -1$$:** The function $$f(x) = 2x + 5$$ is a line with: - Slope = 2 - $$y$$-intercept = 5 Since $$x \leq -1$$, we will: 1. Start at $$x = -1$$ and calculate $$f(-1)$$: $$f(-1) = 2(-1) + 5 = -2 + 5 = 3$$ So the point $$(-1, 3)$$ is on the graph, and this endpoint is included ($$\leq$$). 2. Extend the line to the left ($$x < -1$$) following the slope $$m = 2$$: - For $$x = -2$$: $$f(-2) = 2(-2) + 5 = -4 + 5 = 1$$ This gives the point $$(-2, 1)$$. - For $$x = -3$$: $$f(-3) = 2(-3) + 5 = -6 + 5 = -1$$ This gives the point $$(-3, -1)$$. **Draw this segment** as a solid line starting from $$(-1, 3)$$ and extending leftward through points like $$(-2, 1)$$ and $$(-3, -1)$$. --- ### **Step 2: Analyze the second case $$f(x) = -2$$ for $$x > 5$$:** Here, $$f(x) = -2$$ is a horizontal line where the $$y$$-value is always $$-2$$. 1. This line starts at $$x = 5$$ (but not including 5), so we have an **open circle** at $$(5, -2)$$. 2. Extend the line rightward for $$x > 5$$. --- ### **Step 3: Combine the two pieces:** 1. **First piece**: A solid line for $$x \leq -1$$ starting from $$(-1, 3)$$ and extending leftward with slope 2. 2. **Second piece**: A horizontal line at $$y = -2$$ starting at an open circle at $$(5, -2)$$ and extending rightward. --- Let me know if you’d like a detailed graph sketch or further clarifications! --- ### **Follow-Up Questions:** 1. How do we determine the slope and intercept of a linear function? 2. Why is there an open circle at $$x = 5$$? 3. What would happen if $$x \leq 5$$ was part of the second condition? 4. How do you plot functions with piecewise definitions? 5. How do you determine continuity in a piecewise function like this? **Tip:** For piecewise functions, carefully handle the endpoints (open or closed) and plot each segment accurately!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Graphing Functions
Formulas
y = mx + b (slope-intercept form)
Theorems
Definition of Piecewise Functions
Suitable Grade Level
Grades 9-12
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