# Vertex Form

Vertex Form is a way of expressing a quadratic function in the form:

$$y = a(x-h)^2 + k$$

• $$x$$ represents the dependent variable
• $$y$$ represents the independent variable
• $$a$$ represents the direction the parabola opens and its compression/stretch factor
• $$h$$ represents the factor the parabola gets shifted horizontally
• $$k$$ represents the factor the parabola gets shifted vertically

Vertex Form is most commonly used for finding the vertex of a given quadratic function.
For example, we can identify the vertex of the function $$y = (x - 2)^2 + 5$$ as $$(2, 5)$$ since $$h = 2$$ and $$k = 5$$.

This can be graphically represented as such:

Identify the Vertex Form of a quadratic that has an Axis of Symmetry of $$-9$$, and an Optimal Value of $$6$$.

Identify the Vertex Form of a quadratic that passes through the points (1,5) and (3,29).

## Converting Vertex Form to Standard Form

1. Rearrange the function if necessary so that all terms are on one side
2. Simplify the expression by squaring the binomial
3. Expand the expression using distributive property
4. Further simplify by collecting like terms

Example

Convert $$y = 4(x - 2)^2 + 6$$ to Standard Form.

Since all terms are already on one side, we can skip Step 1.
First, we can simplify the expression by squaring the binomial:

$$y = 4(x - 2)(x - 2) + 6$$
$$y = 4((x)(x) + (x)(-2) + (-2)(x) + (-2)(-2)) + 6$$
$$y = 4(x^2 - 2x - 2x + 4) + 6$$
$$y = 4(x^2 - 4x + 4) + 6$$

Next, we can expand the expression using distributive property:

$$y = (4)(x^2) - (4)(4x) + (4)(4) + 6$$
$$y = 4x^2 - 16x + 16 + 6$$

Finally, further simplify by collecting like terms:

$$y = 4x^2 - 16x + 22$$

Therefore, we can determine that $$y = 4(x - 2)^2 + 6$$ in Standard Form is $$y = 4x^2 - 16x + 22$$.

Convert $$y - 4 = \cfrac{1}{3}(x + 1)^2$$ to Standard Form

## Converting Vertex Form to Factored Form

1. Convert to Standard Form using the steps listed above
2. Factor using various techniques learnt

A rocket travels according to the equation $$h = -4.9(t - 6)^2 + 182$$ where $$h$$ is the height, in metres, above the ground and $$t$$ is the time, in seconds.
1. Sketch a graph of the rocket's motion
2. Find the maximum height of the rocket
3. How long does it take the rocket to reach its maximum height?
4. How high was the rocket above the ground when it was fired?