To determine whether each graph represents a
function, we use the
Vertical Line Test:
>
✔ A graph represents a
function if
no vertical line intersects the graph at more than one point.
>
✘ If any vertical line crosses the graph at two or more points, it is
not a function.
Let’s go through each graph one by one:
---
1)
- This graph is a
curve that opens to the right, like a sideways parabola.
- It fails the vertical line test because some vertical lines (e.g., at $ x = 0 $) intersect the graph at
two points.
-
✘ Not a function
---
2)
- This is a smooth curve that goes up and down (like a wave).
- Every vertical line will intersect the graph at
only one point.
-
✔ It is a function
---
3)
- This is a
parabola opening upward, vertex at origin.
- Vertical line test: every vertical line hits only one point.
-
✔ It is a function
---
4)
- This graph looks like a
sideways S-shape or a cubic-like curve but rotated.
- However, notice that for certain $ x $-values (e.g., around $ x = 1 $), there are
two y-values.
- So, a vertical line would cross the graph twice.
-
✘ Not a function
---
5)
- This is a straight diagonal line from top-left to bottom-right.
- Any vertical line will cross it only once.
-
✔ It is a function
---
6)
- This is an
ellipse (oval shape).
- A vertical line drawn through the center will intersect the ellipse at
two points.
-
✘ Not a function
---
7)
- This graph has two separate curves: one on the left (downward) and one on the right (upward).
- But notice: for some $ x $-values (especially near $ x = -2 $ and $ x = 2 $), there are
two y-values.
- For example, a vertical line at $ x = 0 $ might hit both the upper and lower parts?
- Wait — looking closely: the graph appears to have
two disconnected branches with overlapping $ x $-values.
- Specifically, for $ x = 0 $, there seems to be
two points: one high and one low.
- So yes, it fails the vertical line test.
-
✘ Not a function
---
8)
- This is a
V-shaped graph, possibly absolute value.
- It's continuous, and each $ x $-value maps to exactly one $ y $-value.
- No vertical line hits more than one point.
-
✔ It is a function
---
9)
- This graph shows
three distinct points: $ (-1, 0), (0, 1), (1, 0) $
- Each $ x $-value appears only once.
- Even though it's not connected, since each input has only one output, it satisfies the definition of a function.
-
✔ It is a function
---
✔ Final Answers:
1)
No
2)
Yes
3)
Yes
4)
No
5)
Yes
6)
No
7)
No
8)
Yes
9)
Yes
---
Summary:
| Graph | Is it a Function? |
|-------|-------------------|
| 1 |
✘ No |
| 2 |
✔ Yes |
| 3 |
✔ Yes |
| 4 |
✘ No |
| 5 |
✔ Yes |
| 6 |
✘ No |
| 7 |
✘ No |
| 8 |
✔ Yes |
| 9 |
✔ Yes |
Let me know if you'd like a visual explanation or how to apply this test in general!
Parent Tip: Review the logic above to help your child master the concept of worksheet functions.
