Answer
To determine if \( y \) is a function of \( x \) using the vertical line test, draw vertical lines across the graph. If each line intersects the graph at most once, then \( y \) is a function of \( x \). If any line intersects the graph more than once, \( y \) is not a function of \( x \).
Solution
To determine whether \( y \) is a function of \( x \) using the **vertical line test**, you can follow these steps:
### **Vertical Line Test Explained**
1. **Imagine Vertical Lines:** Visualize or draw vertical lines (parallel to the y-axis) and move them across the entire graph from left to right.
2. **Check Intersections:** Observe how many times each vertical line intersects the graph:
- **If every vertical line intersects the graph at **_most once_**, then **\( y \) is a function of \( x \)**.
- **If any vertical line intersects the graph **_more than once_**, then **\( y \) is not a function of \( x \)**.
### **Why It Works**
A function assigns exactly one output \( y \) for each input \( x \). If a vertical line crosses the graph multiple times at a single \( x \)-value, it means there are multiple \( y \)-values for that \( x \), violating the definition of a function.
### **Example Scenarios**
- **Function Example:** A straight line like \( y = 2x + 3 \). Any vertical line will intersect this line exactly once.
- **Not a Function Example:** A circle, such as \( x^2 + y^2 = 4 \). Vertical lines will intersect the circle twice at most points, meaning multiple \( y \) values for some \( x \) values.
### **Applying to Your Graph**
Since I can't view the specific graph you're referring to, here's how you can apply the test:
1. **Draw or Visualize Vertical Lines:** Starting from the leftmost part of the graph to the rightmost part.
2. **Count Intersections:** For each position of the vertical line:
- **Single Intersection:** Continue checking other positions.
- **Multiple Intersections:** Conclude that \( y \) is not a function of \( x \).
3. **Final Determination:**
- **All Vertical Lines Intersect Once:** \( y \) **is** a function of \( x \).
- **Any Vertical Line Intersects More Than Once:** \( y \) is **not** a function of \( x \).
### **Additional Tips**
- **Algebraic Functions:** Sometimes, you can determine if a relation is a function without graphing, by checking if each \( x \)-value has only one corresponding \( y \)-value.
- **Common Non-Functions:** Circles, horizontal ellipses, and some parabolas (like sideways parabolas) typically fail the vertical line test.
If you can describe the graph or provide specific features (like shapes, lines, curves), I can offer a more tailored explanation!
Reviewed and approved by the UpStudy tutoring team
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