Which of the following functions matches this graph?
On a coordinate plane, a parabola opens up and goes through (negative 1, 3), has a vertex at (0, 0), and goes through (1, 3).
a.
y = x squared
c.
y = negative one-half x squared
b.
y = 3 x squared
d.
y = StartFraction 1 Over 10 EndFraction x squared
 	 	 	 
 


We are given a parabola that opens upwards, has a vertex at (0, 0), and passes through the points (-1, 3) and (1, 3). We need to determine which of the given functions matches this graph.

### Let's examine each option:

1. **Option a: \( y = x^2 \)**
   - This is a standard parabola with a vertex at (0, 0).
   - The points (-1, 1) and (1, 1) are on this parabola, but the given points are (-1, 3) and (1, 3). Therefore, this is not the correct equation.

2. **Option b: \( y = 3x^2 \)**
   - This parabola also has a vertex at (0, 0).
   - For \( x = 1 \) or \( x = -1 \), \( y = 3(1)^2 = 3 \), so the points (-1, 3) and (1, 3) are on this parabola. 
   - This matches the given graph.

3. **Option c: \( y = -\frac{1}{2}x^2 \)**
   - This is a downward-opening parabola due to the negative sign.
   - Since the given parabola opens upwards, this cannot be the correct equation.

4. **Option d: \( y = \frac{1}{10}x^2 \)**
   - This parabola also opens upwards and has a vertex at (0, 0).
   - However, for \( x = 1 \) or \( x = -1 \), \( y = \frac{1}{10}(1)^2 = \frac{1}{10} \), which is much smaller than 3. This is not the correct equation.

### Conclusion:
The correct function is **Option b: \( y = 3x^2 \)**.

Would you like further explanation, or do you have any questions?

Here are some related questions:
1. How do you find the vertex of a parabola?
2. What is the formula for a parabola in vertex form?
3. How can you determine if a parabola opens upwards or downwards?
4. How do you find the equation of a parabola given its vertex and a point?
5. What is the difference between a narrow and a wide parabola?

**Tip**: The coefficient of \( x^2 \) determines how "narrow" or "wide" a parabola is. The larger the coefficient, the narrower the parabola.

Which of the following functions matches this graph? On a coordinate plane, a parabola opens up and goes through (negative 1, 3), has a vertex at (0, 0), and goes through (1, 3).

Identify the quadratic function that matches a given graph with a vertex at (0, 0) and points passing through (-1, 3) and (1, 3). This problem covers key concepts in algebra and graphing parabolas, suitable for students in Grades 9-10.

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