Completing the square is an essential technique in quadratic algebra for transforming quadratic polynomials of the form $ax^2 + bx + c$ into vertex form. This method not only aids in locating the vertex of the parabola but also in sketching its graph and solving optimisation problems.

**Understanding the Process**

Completing the square rewrites a quadratic equation from $y = ax^2 + bx + c$ to $y = p(x + q)^2 + r$, which is vital for identifying the vertex of the parabola.

*Key Concepts:*

- The vertex of the parabola is given by $(-q, r)$.
- The coefficient 'a' affects the parabola's width and direction.
- This technique is useful for solving quadratic equations and graphing quadratic functions.

**Sketching the Graph**

After transforming the quadratic equation into the vertex form, we can sketch its graph. The vertex form gives a clear picture of the parabola's shape and position.

**Steps to Sketch the Graph:**

**1. Identify the Vertex:**$(-q, r)$ give the vertex. This point is the peak or the lowest point of the parabola, depending on the direction of opening

**2. Determine the Direction:**$(p(x + q)^2 + r)$ indicates the direction the parabola opens. If 'p' is positive, the parabola opens upwards; if negative, it opens downwards.

**3. Plot Key Points: **Plot the vertex and a few points on either side of the vertex. Use the symmetry of the parabola about the vertex.

**4. Draw the Parabola:** Join these points with a smooth curve to represent the parabola.

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**Examples: Transforming, Locating the Vertex, and Sketching the Graph**

**Example 1:**

Express $(3x^2 + 9x + 5)$ in complete square form, find its vertex, and sketch the graph.

**Solution:**

Vertex: $\left(-\frac{3}{2}, -\frac{7}{4}\right)$

*Graph Sketching:*

1. Plot the vertex $\left(-\frac{3}{2}, -\frac{7}{4}\right)$.

2. Since $a = 3$, the parabola opens upwards and is narrower than a standard parabola $y = x^2$.

3. Sketch the parabola passing through the vertex.

**Example 2:**

Convert $2x^2 - 4x + 1$ to vertex form, find the vertex, and sketch the graph.

**Solution:**

Vertex: $(1, -1)$

*Graph Sketching:*

1. Plot the vertex $(1, -1)$.

2. With $a = 2$, the parabola opens upwards and is narrower.

3. Draw the parabola through the vertex.

**Example 3:**

Transform $x^2 + 6x + 8$ into vertex form, determine the vertex, and sketch the graph.

**Solution: **

Vertex: $(-3, -1)$

*Graph Sketching:*

1. Mark the vertex $(-3, -1)$.

2. As $a = 1$, the parabola is standard in width and opens upwards.

3. Sketch the parabola passing through the vertex.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.