Math Problem Statement
Which one should i click?
Which one should I click?
Solution
To determine which graph represents , let's analyze the piecewise function:
\begin{cases} -\frac{1}{2}x + 2 & \text{if } x \leq 2 \\ 2x - 7 & \text{if } x > 2 \end{cases}$$ ### Step 1: Analyze the two equations 1. **For $$x \leq 2$$: $$f(x) = -\frac{1}{2}x + 2$$** This is a line with a **negative slope** of $$-\frac{1}{2}$$ and a y-intercept of $$2$$. At $$x = 2$$, the value of the function is: $$f(2) = -\frac{1}{2}(2) + 2 = -1 + 2 = 1.$$ So the line ends at the point $$(2, 1)$$. 2. **For $$x > 2$$: $$f(x) = 2x - 7$$** This is a line with a **positive slope** of $$2$$ and a y-intercept of $$-7$$. At $$x = 2$$, the value of this equation is: $$f(2) = 2(2) - 7 = 4 - 7 = -3.$$ However, the second part starts from $$x > 2$$, so this segment begins slightly to the right of $$x = 2$$. ### Step 2: Graph characteristics - The graph will have two **line segments**: - A **negative-sloped line** on the left up to $$x = 2$$ with a point at $$(2, 1)$$. - A **positive-sloped line** starting at $$x > 2$$. - There will be a **jump discontinuity** at $$x = 2$$ because the function values don’t connect. ### Step 3: Compare graphs - Look for a graph where: 1. The left segment (for $$x \leq 2$$) has a negative slope and ends at $$(2, 1)$$. 2. The right segment (for $$x > 2$$) has a positive slope starting **above** $$x = 2$$, but it doesn’t connect to $$(2, 1)$$. ### Answer: The correct graph is **the one in the bottom-right corner** (pink background). It matches the characteristics of the piecewise function. --- Would you like me to explain any part of this further? Here are 5 related questions to explore: 1. How do you graph a piecewise function step by step? 2. What causes a jump discontinuity in a graph? 3. How can you find the slope and y-intercept of a line? 4. How do you evaluate piecewise functions at a specific $$x$$-value? 5. What are the key differences between continuous and discontinuous functions? **Tip**: Always check for breaks in the graph (discontinuities) when dealing with piecewise functions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Slope and Intercepts
Graph Discontinuity
Formulas
f(x) = -1/2x + 2 when x ≤ 2
f(x) = 2x - 7 when x > 2
Theorems
Piecewise Function Discontinuity
Slope-Intercept Form (y = mx + b)
Suitable Grade Level
Grade 10