When graphing **parabolas** and solving quadratic equations it is often useful to have the function written in a particular form, depending on the context and what characteristics we are interested in.

In all of the above forms, the value of a is the scale factor of the quadratic function, and indicates the direction of opening of the graph. If a>0 then the parabola will open upwards, and if a<0 then the parabola opens downwards. This also means a \neq 0.

If we want to reveal different characteristics of a parabola, we can rewrite the quadratic function in different forms. This can be useful when sketching a graph of a quadratic function where we want to show all the characteristics:

- x-intercepts
- y-intercept
- vertex (absolute maximum or minimum)
- axis of symmetry
- increasing, decreasing and constant intervals
- end-behavior

In addition to this, we can also use the context of a quadratic function to determine if there is an appropriate domain and range, or interpret what the characteristics represent in the context.

Consider the graph.

a

State the coordinates of the vertex of the parabola.

Worked Solution

b

Write the equation of the parabola in vertex form.

Worked Solution

A golf ball is hit into the air and its height h feet above the ground at time t seconds after being hit is given by h = - 16t^{2} + 128t.

a

Assuming the ball starts at a height of 0 feet, determine when it will hit the ground.

Worked Solution

b

Find the greatest height the ball reaches above the ground.

Worked Solution

c

Find the domain constraint for h, so it fits the restrictions of hitting the golf ball. Give your answer using interval notation.

Worked Solution

Sketch the graph of the quadratic function y=2x^2+4x-30, labeling the following key features:

- x-intercepts
- y-intercept
- Vertex coordinates

Consider the quadratic function: y=2x^2+4x-30

a

Rewrite the quadratic equation in a form that allows us to identify the x-intercepts.

Worked Solution

b

Rewrite the quadratic equation in a form that allows us to identify the coordinates of the vertex.

Worked Solution

c

Sketch the graph of the quadratic function, labeling the x- and y-intercepts, and the vertex.

Worked Solution

Idea summary

Quadratic functions could be in the following forms.

**Vertex form**\displaystyle f\left(x\right)=a\left(x-h\right)^2+k\bm{\left(h,k\right)}are the coordinates of the vertex (of the quadratic function)**Factored form**\displaystyle f\left(x\right)=a\left(x-x_1\right)\left(x-x_2\right)\bm{x_1 \text{ and } x_2}are the x-values of the x-intercepts**Standard form**\displaystyle f\left(x\right)=ax^2+bx+c\bm{c}is the y-intercept\bm{x=-\dfrac{b}{2a}}is the equation of the axis of symmetry