To determine which graph represents the function $y = x^2 - x + 1$, we need to analyze the behavior of the quadratic function based on its shape, vertex, and overall characteristics.

Steps to analyze the function $y = x^2 - x + 1$:

1. Form of the Function: The equation is a quadratic function of the form $y = ax^2 + bx + c$, where $a = 1$, $b = -1$, and $c = 1$. Since $a = 1$, the parabola opens upwards.

2. Vertex: The vertex of a parabola given by the general form $y = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex: $x_{\text{vertex}} = \frac{-b}{2a} = \frac{-(-1)}{2(1)} = \frac{1}{2}$ Substituting $x_{\text{vertex}} = \frac{1}{2}$ into the equation to find the y-coordinate: $y_{\text{vertex}} = \left( \frac{1}{2} \right)^2 - \frac{1}{2} + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{3}{4}$ So, the vertex of the parabola is at $\left( \frac{1}{2}, \frac{3}{4} \right)$.

3. Intercepts:

   • Y-intercept: This occurs when $x = 0$: $y = (0)^2 - 0 + 1 = 1$ So, the y-intercept is $(0, 1)$.
   • X-intercepts: The x-intercepts can be found by solving $x^2 - x + 1 = 0$. The discriminant $\Delta$ of this quadratic equation is: $\Delta = (-1)^2 - 4(1)(1) = 1 - 4 = -3$ Since the discriminant is negative, there are no real x-intercepts. This means the parabola does not cross the x-axis.

4. Shape of the Graph: The graph will be an upward-facing parabola with its vertex at $\left( \frac{1}{2}, \frac{3}{4} \right)$, a y-intercept at $(0, 1)$, and no x-intercepts.

Based on these observations, the correct graph in Figure 1 should:

• Be an upward-facing parabola.
• Have a vertex around $\left( \frac{1}{2}, \frac{3}{4} \right)$.
• Pass through the point $(0, 1)$.
• Not intersect the x-axis.

Would you like me to explain more about any of these steps? If you have the figure available, you can also upload it for direct comparison.

Here are five related questions you could consider:

1. How do you find the axis of symmetry for a quadratic function?
2. What is the effect of the coefficient $a$ on the width and direction of a parabola?
3. How can you use the discriminant to determine the number of x-intercepts?
4. How do you determine the range of a quadratic function?
5. How does the vertex form of a quadratic equation help in graphing the function?

Tip: If you're unsure whether a quadratic function has real x-intercepts, always check the discriminant $\Delta = b^2 - 4ac$.