Consider the graph below, which shows the vertical position, y in feet, of a water balloon thrown by a child from the low diving board of a pool over time, x in seconds:

The function representing the projectile motion of the water balloon is y=-3x^2+3x+6.

- Shorena says that the function y=-3(x+1)(x-2) is equivalent to the given function. How can we determine if she is correct?
- How do the characteristics of the graph relate to the context?
- How might the function y=-3(x+1)(x-2) relate to the graph?

One way to represent quadratic functions is using the **factored form**. This form allows us to identify the x-intercepts, the direction of opening, and scale factor of the quadratic function.

\displaystyle f(x)=a(x-x_1)(x-x_2)

\bm{x_1, \,x_2}

x-values of the x-intercepts

\bm{a}

scale factor

If a>0, then the quadratic function opens upwards and has a minimum value.

If a<0 then the quadratic function opens downwards and has a maximum value.

The x-intercepts are the points where f(x)=0, so we refer to x_1 and x_2 as the **zeros** of the function.

To draw the graph of a quadratic function, we generally want to find three different points on the graph, such as the x- and y-intercepts.

Consider the graph of a quadratic function:

a

Identify the coordinates of the x- and y-intercepts of the function.

Worked Solution

b

Find the equation of the quadratic function in factored form.

Worked Solution

Consider the quadratic function:

y=2x^{2} + 4x - 48

a

State the coordinates of the x-intercepts.

Worked Solution

b

Determine the coordinates of the y-intercept.

Worked Solution

c

Determine the coordinates of the vertex.

Worked Solution

d

Draw the graph of the function.

Worked Solution

Identify the characteristics of h\left(x\right)=\dfrac{1}{6}\left(3x + 2\right)\left(x - 7\right).

a

Identify the factors of the function.

Worked Solution

b

Identify the roots of the function.

Worked Solution

c

Identify the zeros of the function.

Worked Solution

d

State the x-intercepts of the function.

Worked Solution

The graph of a quadratic function has x-intercepts at \left(-2,\,0\right) and \left(1,\,0\right) and passes through the point \left(-3,\,-2\right). Write an equation in factored form that models this quadratic.

Worked Solution

Find the equation that models the graph shown below.

Worked Solution

A cannonball is fired from the edge of a cliff which is 15 meters above sea level. The peak of the cannonball's arc is 20 meters above sea level and 10 meters horizontally from the cliff edge. The cannonball lands in the sea 30 meters away from the base of the cliff.

The path of the cannonball is shown on the following graph, but the axes have not been labeled.

a

Label the axes of the graph to match the information provided.

Worked Solution

b

Determine the factored equation which models the path of the cannonball.

Worked Solution

c

A second cannonball is fired, and this one can be modeled by the equation: y=-\frac{1}{15}\left(x+12\right)\left(x-27\right)Use this model to predict where the cannonball landed.

Worked Solution

Idea summary

To write the equation of the graph of a quadratic function in factored form, substitute the x-intercepts for x_1 and x_2 in the equation y=a(x-x_1)(x-x_2), then use any other point on the graph to substitute for x and y and solve for a, the scale factor.