what are 3 different quadratic equations with a vertex at (2,5)

To write three different quadratic equations with a vertex at \((2,5)\), we can use the **vertex form** of a quadratic equation:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola, and \(a\) controls the "width" and direction of the parabola (whether it opens upwards or downwards).

Given the vertex \((2, 5)\), the equation becomes:

\[
y = a(x - 2)^2 + 5
\]

Now, we can create three different quadratic equations by assigning different values to \(a\):

1. **For \(a = 1\):**
   \[
   y = (x - 2)^2 + 5 \quad \text{or} \quad y = x^2 - 4x + 9
   \]

2. **For \(a = -2\):**
   \[
   y = -2(x - 2)^2 + 5 \quad \text{or} \quad y = -2x^2 + 8x - 3
   \]

3. **For \(a = \frac{1}{2}\):**
   \[
   y = \frac{1}{2}(x - 2)^2 + 5 \quad \text{or} \quad y = \frac{1}{2}x^2 - 2x + 7
   \]

These are three different quadratic equations, all with a vertex at \((2, 5)\).

Would you like further details or have any questions?

Here are five related questions:
1. How does the value of \(a\) affect the shape of the parabola?
2. What is the difference between vertex form and standard form of a quadratic equation?
3. How can you convert a quadratic equation from vertex form to factored form?
4. How do you find the axis of symmetry for a quadratic function?
5. How does the value of the constant \(k\) affect the position of the parabola?

**Tip:** The sign and value of \(a\) in the quadratic equation determine the direction and width of the parabola.

What are 3 different quadratic equations with a vertex at (2,5)?

Discover three different quadratic equations that share the vertex (2,5). This guide covers the vertex form of quadratic equations and how varying the coefficient affects the graph. Suitable for students in Grades 7-9.

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