Graphing Quadratic Functions

1. The Anatomy of a Parabola

The graph of a quadratic function, y = ax² + bx + c, is a symmetrical U-shaped or n-shaped curve called a parabola. To sketch it accurately, we need to identify its key features.

Here are the features we will be finding:

• Shape (or Orientation): Is the parabola U-shaped ("opens up") or n-shaped ("opens down")?
• y-intercept: The point where the graph crosses the vertical y-axis.
• x-intercepts (Roots): The point(s) where the graph crosses the horizontal x-axis. A parabola can have two roots, one root, or no real roots.
• Vertex (or Turning Point): The minimum or maximum point of the parabola.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

Let's begin by watching a video that provides an excellent overview of these features and how to identify them from a graph.

Now, let's learn how to find these features algebraically from the function's equation.

2. Finding the Key Features Algebraically

Let's consider a quadratic function in its general form: y = ax² + bx + c.

Shape (Orientation)

This is the easiest feature to determine. It depends entirely on the sign of the coefficient a (the number in front of x²).

• If a > 0 (positive), the parabola is U-shaped.
• If a < 0 (negative), the parabola is n-shaped.

This is because a negative a value corresponds to a reflection in the x-axis, as we saw in our previous lesson.

y-intercept

The y-intercept occurs where x = 0. If we substitute x = 0 into the equation:
y = a(0)² + b(0) + c = c
So, the y-intercept is always at the point (0, c).

x-intercepts (Roots)

The x-intercepts occur where y = 0. To find them, you need to solve the quadratic equation ax² + bx + c = 0. You can use any of the methods we've previously covered:

• Factorising (factoring)
• Completing the square
• The quadratic formula

Remember, the value of the discriminant (b² - 4ac) will tell you how many real roots the function has.

3. Finding the Vertex and Axis of Symmetry

Finding the vertex is the most important step in graphing a parabola. We'll explore two highly effective methods.

Method 1: Completing the Square (The Transformation Method)

This method directly connects to our previous lessons on transformations. By completing the square, we can convert the general form y = ax² + bx + c into the vertex form:
$y=a(x-h{)}^2+k$
When the equation is in this form, you can see it as a series of transformations applied to y = x²:

1. A horizontal shift of h units.
2. A vertical stretch by a factor of a.
3. A vertical shift of k units.

The result is that the vertex of the parabola is located at the point (h, k). The axis of symmetry is the vertical line x = h.

The following video masterfully demonstrates this process.

Method 2: Using a Formula (The Direct Method)

A more direct way to find the vertex is by using a formula for its x-coordinate. For any quadratic y = ax² + bx + c, the x-coordinate of the vertex is given by:
$x=-\frac{b}{2a}$
This formula is incredibly useful. It directly gives you the equation for the axis of symmetry, x = -b / (2a).

To find the y-coordinate of the vertex, you simply substitute this x-value back into the original quadratic equation.

The resource below explains this formula. It uses American terminology, where "general form" is ax²+bx+c and "standard form" is the vertex form we saw above.

A fascinating insight is that the x-coordinate of the vertex is always exactly halfway between the two roots. The formula x = -b / (2a) is simply a way of finding the midpoint of the roots as given by the quadratic formula.

4. Putting It All Together: Sketching the Graph

We now have all the tools we need. Here is a step-by-step guide to sketching a quadratic graph.

Let's use the example y = x² + 2x - 3.

1. Shape: The coefficient of x² is a = 1, which is positive. So, the graph is U-shaped.

2. y-intercept: The constant term is c = -3. So, the y-intercept is at (0, -3).

3. Vertex: Let's use the formula x = -b / (2a). Here a = 1, b = 2.

   • x-coordinate: x = -2 / (2 * 1) = -1.
   • y-coordinate: y = (-1)² + 2(-1) - 3 = 1 - 2 - 3 = -4.
   • The vertex is at (-1, -4).

4. Axis of Symmetry: This is the vertical line through the vertex. Its equation is x = -1.

5. x-intercepts (Roots): Set y = 0 and solve x² + 2x - 3 = 0.

   • We can factorise this: (x + 3)(x - 1) = 0.
   • The roots are at x = -3 and x = 1. The coordinates are (-3, 0) and (1, 0).

6. Sketch: Plot the key points you've found:

   • Vertex: (-1, -4)
   • Intercepts: (0, -3), (-3, 0), (1, 0)
   • Draw a smooth, symmetrical U-shaped curve passing through these points. You can also use the axis of symmetry to find a "bonus" point. Since (0, -3) is 1 unit to the right of the axis x = -1, there must be a symmetric point 1 unit to the left, at (-2, -3).

The webpage from Third Space Learning provides another excellent worked example.

Conclusion

You now know how to deconstruct and graph any quadratic function by focusing on its essential characteristics. This "key features" approach is far more insightful than just plotting points, as it reveals the underlying structure of the function, which is directly linked to the transformations we've been studying.

Key Takeaways:

• A parabola's graph is defined by its shape (orientation), vertex, axis of symmetry, and intercepts.
• The sign of the a term in y = ax² + bx + c determines if the parabola is U-shaped (a>0) or n-shaped (a<0).
• The y-intercept is (0, c), and the x-intercepts (roots) are found by solving ax² + bx + c = 0.
• The vertex can be found in two main ways:
  1. Completing the square to get vertex form y = a(x - h)² + k, where the vertex is (h, k).
  2. Using the formula x = -b / (2a) to find the x-coordinate, then substituting to find the y-coordinate.
• The axis of symmetry is the vertical line x = h that passes through the vertex.

Preview of the Next Lesson:
We will continue to practice this method of analysing and graphing functions based on their key features. In the next lesson, we will apply these skills to two new function families: absolute value functions and basic square root functions. You'll see that the same fundamental principles of identifying key points and understanding transformations apply there as well.