Finding the Vertex by Completing the Square (3.15.1)

Question 1

• Standard form of a quadratic function: f (x) = ax^2+ bx + c.

• Vertex of a parabola: (h, k).
• h = –b/2a.
• k = f (h).

• Vertex form of a quadratic function: f (x) = a(x – h)^2 + k.

Answer 1

• Standard form of a quadratic function: f (x) = ax^2+ bx + c.

• Vertex of a parabola: (h, k).
• h = –b/2a.
• k = f (h).

• Vertex form of a quadratic function: f (x) = a(x – h)^2 + k.

Question 2

• Completing the square is one method for changing a quadratic function in standard form into vertex form in order to easily identify the vertex of the parabola.

Answer 2

• Completing the square is one method for changing a quadratic function in standard form into vertex form in order to easily identify the vertex of the parabola.

Question 3

In this example the x-terms are (x^2+ 4x).

To complete that square, divide the 4 in half, square it, and
add it inside the x-term expression so that you have
(x^2 + 4x + 4). To maintain the balance of the function, add
a –4 outside the x-term expression.

Combine like terms and contract the square.
The vertex is at (–2, –3).

In addition you know this will be a positive parabola.
Your graphing job is easy.

Follow the same process for this example as above.
Within the x-expression, you’ll add 16.

Outside that expression, add –16 for balance.
Factor into squared form and your vertex is (4, –11).

It’s a positive parabola.

Again, follow the same process as above.
Calculate the value to add and subtract.
Factor into squared form.
Graph the vertex.

Note that this is a negative parabola and graph the curve.

Answer 3

In this example the x-terms are (x^2+ 4x).

To complete that square, divide the 4 in half, square it, and
add it inside the x-term expression so that you have
(x^2 + 4x + 4). To maintain the balance of the function, add
a –4 outside the x-term expression.

Combine like terms and contract the square.
The vertex is at (–2, –3).

In addition you know this will be a positive parabola.
Your graphing job is easy.

Follow the same process for this example as above.
Within the x-expression, you’ll add 16.

Outside that expression, add –16 for balance.
Factor into squared form and your vertex is (4, –11).

It’s a positive parabola.

Again, follow the same process as above.
Calculate the value to add and subtract.
Factor into squared form.
Graph the vertex.

Note that this is a negative parabola and graph the curve.