To determine whether each graph represents a
function, we use the
Vertical Line Test:
>
Vertical Line Test: If any vertical line intersects the graph at more than one point, then the graph does
not represent a function. If every vertical line intersects the graph at most once, then it
does represent a function.
Let’s go through each graph one by one:
---
1)
- The graph is a
curve that opens to the right, like a sideways parabola.
- A vertical line (e.g., at $ x = 0 $) will intersect the graph at
two points.
-
✔ Not a function
---
2)
- This is a smooth curve with peaks and valleys (like a wave).
- Any vertical line will cross the graph at only
one point.
-
✔ It is a function
---
3)
- This is a
parabola opening upward, vertex at the bottom.
- Every vertical line crosses the graph at only
one point.
-
✔ It is a function
---
4)
- This graph looks like a
sideways "S" shape, possibly part of a cubic or other relation.
- At certain $ x $-values (e.g., near $ x = 0 $), a vertical line would pass through
two points.
-
✔ Not a function
---
5)
- This is a
straight diagonal line from top-left to bottom-right.
- A vertical line will intersect it at exactly
one point.
-
✔ It is a function
---
6)
- This is an
ellipse (oval shape), centered around the origin.
- A vertical line through the center (e.g., $ x = 0 $) will intersect the graph at
two points (top and bottom).
-
✔ Not a function
---
7)
- This graph has
two separate curves: one on the left and one on the right.
- For some $ x $-values (e.g., $ x = -1 $), there are two corresponding $ y $-values (one on each branch).
-
✔ Not a function
---
8)
- This is a
piecewise linear graph with a sharp corner (like a V-shape but shifted).
- Each vertical line hits the graph at
only one point.
-
✔ It is a function
---
9)
- This is a
vertical line segment from $ y = 0 $ to $ y = 20 $ at $ x = 10 $.
- A vertical line at $ x = 10 $ intersects the graph at
many points (all along the segment).
-
✔ Not a function
---
✔ Final Answers:
1)
No
2)
Yes
3)
Yes
4)
No
5)
Yes
6)
No
7)
No
8)
Yes
9)
No
---
📌 Summary:
A graph represents a
function if and only if
each input ($ x $) corresponds to
exactly one output ($ y $). The
vertical line test helps us check this visually.
Let me know if you'd like these answers filled in on the worksheet!
Parent Tip: Review the logic above to help your child master the concept of identifying graphs of functions worksheet.
