Standard 7: Parabolas

Standard 7: Analyze and graph quadratic functions using the technique of completing the square to find the vertex and axis of symmetry

Parent Graph Toolkit for Parabolas (click to make it larger):

Important terms to understand:

• Quadratic function
• Standard form
• Graphing form
• Stretch factor
• Perfect square trinomial
• Completing the square
• Vertex
• Axis of symmetry

Quadratic function: Any polynomial in which the highest power of x is 2. These are also known as parabolas.
Examples: x² + 2x + 4, 3x² - 4, 2(x + 4)² - 2

Standard form: Standard form of a quadratic function is y = ax² + bx + c, where a, b, and c are real numbers.
Examples: x² + 3x + 3, 1.3x² - 2.5x + 7

Graphing form: Graphing form of a quadratic function is y = a(x - h)² + k.
Examples: 3(x - 1)² + 4, 0.2(x + 3)² - 5

Stretch factor: The stretch factor of a parabola controls the shape of the parabola. It is the a in graphing form.

When a > 1, the parabola will have a vertical stretch (it will look taller and thinner than usual).
When a = 1, the parabola will have no change in shape.
When 0 < a < 1, the parabola will have a vertical compression (it will look shorter and wider than usual).

When a < -1, the parabola will have a vertical stretch and be flipped horizontally (it will look thinner and taller and be flipped upside-down).
When a = -1, the parabola will simply be flipped upside down.
When -1 < a < 0, the parabola will have a vertical compression and be flipped horizontally (it will look shorter and wider and be flipped upside-down).

When a = 0, then we no longer have a parabola, because the (x - h)² goes away. So, we do not consider this case.

Perfect square trinomial: A perfect square trinomial is any quadratic function that can be factored into the form (x + a)², where a is a number.
Examples: x² + 2x + 1 = (x + 1)², x² - 6x + 9 = (x - 3)²

For any perfect square trinomial, we have the identity

(x + a)² = x² + (2a)x + a²

Completing the square: Completing the square is a process we use to turn a parabola from standard form into graphing form. As an example, we will look at y = x² + 4x + 9.

We want to find a perfect square trinomial that looks mostly like this by matching the first two terms, x² + 4x.

This perfect square trinomial will be (x + 2)² = x² + 4x + 4. We found this by using our identity of perfect square trinomials from above. In this case, (2a) = 4, so a = 2

So, our perfect square trinomial is (x + 2)² = x² + 4x + 4, but we need to make it look like the original function, y = x² + 4x + 9:

(x + 2)² = x² + 4x + 4 + 5 = x² + 4x + 9.

So, y = (x + 2)² + 5.

Vertex: The vertex of a parabola is the lowest point when it opens up, and it is the highest point when it opens down. The x-coordinate is always in the middle of the x-intercepts (roots) of the parabola.

When we have a parabola in graphing form, the vertex is always at (h, k). For example, in y = 2(x - 4)² + 3, the vertex is at (4, 3).

Axis of symmetry: The axis of symmetry for a parabola is the vertical line that splits the parabola in half. This line always passes through the vertex of the parabola, so if we know the vertex, (h, k), the equation of the axis of symmetry is always x = h.


Homework assignments:

#1

PG 13, 15, 21, 23, 24, 27, 28, 38, 39, 144, 163

Due Date: 14 Feb 2012

Related videos:

These videos explain the process a little differently than the way I showed above and in class, but it may help you to see it a different way.

Khan Academy video on completing the square

Just Math Tutorials video on completing the square