To solve this without a calculator, let's approach it step-by-step. The function given is:

$y = (x - 1)^2$

This is a basic quadratic function in the form of $y = (x - h)^2 + k$, which represents a parabola that opens upwards. Here, $h = 1$ and $k = 0$, so the parabola is shifted 1 unit to the right along the x-axis and has its vertex at $(1, 0)$.

Step 1: Identify Key Points

1. Vertex: The vertex is the point where the parabola changes direction. Since $h = 1$ and $k = 0$, the vertex of this function is at: $(1, 0)$

2. Axis of Symmetry: The axis of symmetry for this parabola is the vertical line that passes through the vertex. For our function, this line is: $x = 1$

3. Other Points: To graph this without a calculator, choose values of $x$ around the vertex to find corresponding $y$-values.

   Let's plug in some values for $x$ near 1:

   • For $x = 0$: $y = (0 - 1)^2 = 1$ So, the point is $(0, 1)$.

   • For $x = 2$: $y = (2 - 1)^2 = 1$ So, the point is $(2, 1)$.

   • For $x = -1$: $y = (-1 - 1)^2 = 4$ So, the point is $(-1, 4)$.

   • For $x = 3$: $y = (3 - 1)^2 = 4$ So, the point is $(3, 4)$.

   Now we have a set of points: $(1, 0)$, $(0, 1)$, $(2, 1)$, $(-1, 4)$, and $(3, 4)$.

Step 2: Plot the Points and Draw the Parabola

1. Start by plotting the vertex $(1, 0)$ on the graph.
2. Then, plot the points we calculated: $(0, 1)$, $(2, 1)$, $(-1, 4)$, and $(3, 4)$.
3. Since this is a parabola, connect these points with a smooth curve opening upwards.

Summary of Steps

• Identify the vertex at $(1, 0)$.
• Plot points around the vertex.
• Draw the parabola through these points.

Would you like more detailed information on plotting these points or understanding the transformations? Here are some follow-up questions you might find helpful:

1. How does shifting the vertex affect the graph of a parabola?
2. What is the significance of the axis of symmetry in a parabola?
3. How would the graph change if the function were $y = -(x - 1)^2$?
4. Why is $y = x^2$ the parent function for all quadratic equations?
5. How do you determine if a parabola opens upward or downward?

Tip: To check your points, substitute values into the equation step-by-step to verify accuracy.