# Standard Form

## Learning goals

1. Completing the squares

3. Word problems/tips

## Summary of the unit

The standard form of a quadratic function is a little different from the general form. The standard form makes it easier to graph. Standard form looks like this: f (x) = a(x - h)2 + k , where a≠ 0 . In standard form, h = - and k = c - . The point (h, k) is called the vertex of the parabola. The line x = h is called the axis of the parabola. A parabola is symmetrical with respect to its axis. The value of the function at h = k . If a < 0 , then k is the maximum value of the function. If a > 0 , then k is the minimum value of the function.

## Completing the Square

Not all quadratic functions can be easily factored. Another method, called completing the square, makes it easier to factor a quadratic function. When a = 1 , a quadratic function f (x) = x 2 + bx + c = 0 can be rewritten x 2 + bx = c . Then, by adding ()2 to both sides, the left side can be factored and rewritten (x + )2 . Taking the square root of both sides and subtracting from both sides solves for the roots.

## Steps

Step 1: Group the x^2 and the x terms together.

Step 2: Common factor only the constant terms, if posssible.

Step 3: Complete the square.

Step 4: Write it as a binomial squared.

Example: y= 4x^2 + 16x + 3 y= (4x^2 + 16x) + 3 y= 4(x^2 + 4x) + 3 y= 4(x^2 + 4x + 4 - 4) + 3

-> There needs to be one positive and one negative number to balance each other out

4 ^2 2 y= 4[(x^2 + 4x + 4) - 4] + 3 y= 4 (x+2)^2 - 4 + 3 y= 4 (x+2)^2 - 1

The vertex is (-2, -1)

## Extra Help

Completing the square

What is it?

To find the x-intercepts/zeroes in an equation that cannot be factored, you can use the quadratic formula

In order to use this formula, your equation needs to be in standard form - y=ax^2 +bx + c

Substitute the a, b, and c values into the formula to get your answer.

The "One Direction" Quadratic Formula Song
Example 1

3x^2 - 14x - 5

a = 3, b = -14, c = -5

x = 14 -14^2 - 4(3)(-5)

2(3)

= 14 256

The exact roots are

14+ 256 and 14 - 256

6 6

The approximate roots are 16.67 and 11.33 In the case of a negative number inside the square root, the solution is not possible because you cannot square root a negative number.

## Word Problems Tips/Hints

When the question is asking for the vertex - complete the square.

To find the x-intercepts - use the quadratic formula.

Example 1

A garden measuring 12 metres by 16 metres is to have a pedestrian pathway installed all around it, increasing the total area to 285 square metres. What will be the width of the pathway?

Example 2

The height of a coin in the air t seconds after it is flipped can be modeled by y = -16t^2 + 32t + 5, where t and y are measured in m. What is the maximum height that the coin reaches?

->To solve this, we need to complete the square because it is asking for the maximum height (y value of the vertex). y=(-16t^2 + 32t) + 5 y= -16(t^2 - 2) + 5 y= -16 (t^2 - 2 + 1) -1) +5 -> -16 multiplies with -1 y= -16 (t-1)^2 + 21

The vertex is (1,21).

Therefore the maximum height of the coin is 21m.