The equation you provided is \( Y = x^2 - 2x + 2 \). This is a quadratic function. Let's analyze its key features:

### 1. **Standard Form**: 
The function is already in standard form \( Y = ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -2 \)
- \( c = 2 \)

### 2. **Vertex**:
The vertex of a quadratic function in standard form can be found using the formula:
\[
x_{\text{vertex}} = \frac{-b}{2a}
\]
Substituting the values:
\[
x_{\text{vertex}} = \frac{-(-2)}{2(1)} = 1
\]
To find the y-coordinate of the vertex, substitute \( x = 1 \) into the equation:
\[
Y_{\text{vertex}} = (1)^2 - 2(1) + 2 = 1 - 2 + 2 = 1
\]
Thus, the vertex is \( (1, 1) \).

### 3. **Axis of Symmetry**:
The axis of symmetry is the vertical line that passes through the vertex:
\[
x = 1
\]

### 4. **Y-Intercept**:
The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:
\[
Y = (0)^2 - 2(0) + 2 = 2
\]
So, the y-intercept is \( (0, 2) \).

### 5. **Discriminant**:
The discriminant of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
\Delta = b^2 - 4ac
\]
Substituting the values:
\[
\Delta = (-2)^2 - 4(1)(2) = 4 - 8 = -4
\]
Since the discriminant is negative, this quadratic function does not have real roots. This means the graph does not cross the x-axis and has no real x-intercepts.

### Graph:
The graph is a parabola that opens upwards because \( a > 0 \). It has a vertex at \( (1, 1) \), and it does not intersect the x-axis.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you determine whether a parabola opens upwards or downwards?
2. How would the equation change if the parabola had real roots?
3. What does the discriminant tell you about the nature of the roots?
4. How would you convert the given quadratic equation into vertex form?
5. How would the vertex change if the equation had a different \( b \) value?

**Tip**: To quickly find the vertex of a parabola, use the formula \( x = -\frac{b}{2a} \).

Discover how to solve and analyze the quadratic equation Y = x² - 2x + 2, focusing on key algebraic concepts such as the vertex, axis of symmetry, and discriminant. This guide includes a step-by-step explanation and is ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation