Problem 34 - Introductory Algebra for College Students

Question

Graph the parabola whose equation is given $$y=x^{2}-2 x+1$$

Step-by-Step Solution

Answer

The equation \(y=x^2-2x+1\) can be rewritten in standard form as \(y=(x-1)^2\). The vertex is (1,0) and the axis of symmetry is \(x=1\). The graph forms a upward opening parabola through these points.

Step 1: Convert to standard form

Write the given quadratic equation in standard form \(y=a(x-h)^2+k\). To do this, we complete the square. Rewrite the given equation as \(y = (x-1)^2\).

Step 2: Identify the vertex

The vertex of a parabola \(y=a(x-h)^2+k\) is \((h , k)\). From the standard form equation \(y = (x-1)^2\), the vertex is \((1, 0)\).

Step 3: Identify the Axis of Symmetry

The axis of symmetry for a parabola \(y=a(x-h)^2+k\) is \(x = h\). So, for the standard form equation \(y = (x-1)^2\), the axis of symmetry is \(x = 1\).

Step 4: Plot the graph

Plot the vertex, and two other points either side of the axis of symmetry, then draw a smooth curve through these points to complete the graph.

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is essential for graphing parabolas. The standard form is expressed as
\(y = ax^2 + bx + c\).
Each letter represents a specific value:

• \(a\) affects the direction and width of the parabola.
• \(b\) influences the location of the vertex and the axis of symmetry.
• \(c\) represents the y-intercept, the point where the parabola crosses the y-axis.

Completing the Square

To graph a parabola accurately, we often convert the standard form to vertex form,
\(y = a(x-h)^{2} + k\),
by completing the square. This process creates a perfect square trinomial from the original quadratic equation, allowing us to easily identify the vertex and plot the parabola. In our exercise, converting\(y = x^{2} - 2x + 1\)
to vertex form results in\(y = (x-1)^{2}\),
simplifying the graphing process.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point of the curve, depending on its direction. It's where the parabola changes direction. In vertex form,
\(y = a(x-h)^{2} + k\),
the vertex is
(\(h, k\)).
It is crucial for graphing because it gives us a starting point from which we can create a symmetric curve. In our example, the vertex form of the equation is
\(y = (x-1)^{2}\)
which shows that the vertex is at
(1, 0).

Importance of the Vertex

Identifying the vertex helps in sketching the parabola accurately and understanding its behavior. It's the turning point and therefore critical in determining the minimum or maximum value of the quadratic function.

Axis of Symmetry

Every parabola has an axis of symmetry, a vertical line that divides it into two mirror-image halves. For the equation
\(y = a(x-h)^{2} + k\),
the axis of symmetry is always
\(x = h\).
This line passes through the vertex, emphasizing the parabola’s symmetrical nature. In the example we're exploring, with our parabola's vertex form equation as
\(y = (x-1)^{2}\),
the axis of symmetry is
\(x = 1\).

Role in Graphing

The axis of symmetry helps us graph parabolas by ensuring that points are reflected evenly on both sides of the parabola. During plotting, after drawing the vertex, you can pick points on one side of the axis and reflect them across to find corresponding points on the other side, facilitating a balanced curve.