Pre-Calculus 11 Ch. 3 Quadratic Functions. Name:__________________

Lesson Notes 3.2: Investigating Quadratic Functions in Standard Form

Objectives:

• identifying quadratic functions in standard form

• determining the vertex, domain and range, axis of symmetry, maximum or minimum

value, and x-intercepts and y-intercept for quadratic functions in standard form

• graphing and analysing quadratic functions in applied situations

When a player kicks or punts a football into the air, it reaches a maximum height before falling

back to the ground. The moment it leaves the punter’s foot to the moment it is caught or hits the ground

is called the hang time of the punt. A punter attempts to kick the football so there is a longer hang time

to allow teammates to run downfield to tackle an opponent who catches the ball. The punter may think

about exactly where or how far downfield the football will land. How can you mathematically model the

path of a football through the air after it is punted?

The path of a football through the air is just one of many real-life phenomena that can be

represented by a quadratic function. A quadratic function of the form f (x) = ax2 + bx + c is written

in standard form.

Example 1) Which functions are quadratic? Explain.

a) f(x) = 5x2 – 6x

b) f(x) = 8 + x

3

c) f(x) = 2x(x – 1)(3x + 4)

d) f(x) = (4x + 1)( –2x + 3)

Example 2) For each graph of a quadratic function, identify the following:

a) i) the direction of opening

ii) the coordinates of the vertex

iii) the maximum or minimum value

iv) the equation of the axis of symmetry

v) the x-intercepts and y-intercept

vi) the domain and range

b) i) the direction of opening

ii) the coordinates of the vertex

iii) the maximum or minimum value

iv) the equation of the axis of symmetry

v) the x-intercepts and y-intercept

vi) the domain and range

c) i) the direction of opening

ii) the coordinates of the vertex

iii) the maximum or minimum value

iv) the equation of the axis of symmetry

v) the x-intercepts and y-intercept

vi) the domain and range

Example 3) The x-coordinate of the vertex is given by x = a

b

2

−

. Use this information to determine

the vertex of each quadratic function.

a) y = x2 + 6x + 2

b) y = 3x2 – 12x + 5

Example 3) A frog sitting on a rock jumps into a pond. The height, h, in cm, of the frog above the

surface of the water as a function of time, t, in seconds, since it jumped can be modeled by the

function h(t) = –490t2 + 150t + 25. Where appropriate, answer the following questions to the

nearest tenth.

a) Graph the function.

b) What is the y-intercept? What does it represent in this situation?

c) What maximum height does the frog reach? When does it reach that height?

d) When does the frog hit the surface of the water?

e) What are the domain and range in this situation?

f) How high is the frog 0.25 s after it jumps?

Example 4) A rancher has 100 m of fencing available to build a rectangular corral.

a) Write a quadratic function in standard form to represent the area of the corral.

b) What are the coordinates of the vertex? What does the vertex represent in this situation?

c) Sketch the graph for the function you determined in part a).

d) Determine the domain and range for this situation.

e) Identify any assumptions you made in modeling this situation mathematically.