Y=x²-4x+6 find x and y intercept. Find out vertices of the parabola.

To solve for the x-intercept, y-intercept, and vertices of the parabola given by the equation \( y = x^2 - 4x + 6 \), follow these steps:

### 1. Finding the x-intercept
The x-intercept occurs where \( y = 0 \). Set the equation equal to zero and solve for \( x \):
\[ x^2 - 4x + 6 = 0 \]

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \). Use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -4 \), and \( c = 6 \).

Calculate the discriminant:
\[ \Delta = b^2 - 4ac = (-4)^2 - 4(1)(6) = 16 - 24 = -8 \]

Since the discriminant is negative, there are no real x-intercepts. The parabola does not cross the x-axis.

### 2. Finding the y-intercept
The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:
\[ y = 0^2 - 4(0) + 6 = 6 \]

So, the y-intercept is:
\[ (0, 6) \]

### 3. Finding the vertex of the parabola
The vertex of a parabola given by \( y = ax^2 + bx + c \) is found at \( x = -\frac{b}{2a} \).

Calculate the x-coordinate of the vertex:
\[ x = -\frac{-4}{2(1)} = \frac{4}{2} = 2 \]

Substitute \( x = 2 \) back into the equation to find the y-coordinate:
\[ y = (2)^2 - 4(2) + 6 = 4 - 8 + 6 = 2 \]

So, the vertex is:
\[ (2, 2) \]

### Summary:
- **x-intercept:** No real intercepts (the parabola does not cross the x-axis).
- **y-intercept:** \( (0, 6) \)
- **Vertex:** \( (2, 2) \)

Would you like any more details or have any questions?

Here are 5 more questions you might want to explore:
1. How do you complete the square for a quadratic equation?
2. How do you find the axis of symmetry for a parabola?
3. How do you determine if a quadratic function opens upwards or downwards?
4. What are the properties of the vertex form of a quadratic equation?
5. How can you graph a quadratic function using its vertex and intercepts?

**Tip:** When solving quadratic equations, always check the discriminant (\( b^2 - 4ac \)) first to determine the nature of the roots.

Learn how to find the x-intercept, y-intercept, and vertex of the parabola defined by the equation y = x² - 4x + 6. This problem involves using the quadratic formula and understanding the properties of quadratic functions. Suitable for high school students.

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