OPT 4: Day 03 Exploring Quadratic Functions Day 2 of 2

Up and Down or Down and Up - Day 2 of 2

Exploring Quadratic Functions

TOPIC 1
Introduction to Quadratic
Functions

Up and Down or
Down and Up

Endless Forms
Most Beautiful

More Than Meets the
Eye

You Lose Some,
You Lose Some

1

2

3

4

Hotspot

Where are the roots located?

Audio Response

Does this function open up or down? Eplain your answer.

$$y=-2x^2+7x\ +\ 3$$

Open Audio Recorder

Learning Goals

❏

Write quadratic functions to model contexts.

❏

Graph quadratic functions using technology.

❏

Interpret the key features of quadratic functions in terms of a context.

❏

Identify the domain and range of quadratic functions and their contexts.

parabola

vertical motion

model

root

KEY TERMS

The workbook
assignment IS
DUE TODAY!

Activity 3

Using a Quadratic Function to Model Vertical Motion

You can model the motion of a pumpkin released from a catapult using a vertical motion model. A vertical motion model is a quadratic equation that models the height of an object at a given time.

> Consider the equation for a vertical motion model.

y = -16t² + v_ot + h_o

In this equation,

y represents the height of the object in feet,

t represents the time in seconds that the object has been moving,

v_o represents the initial vertical velocity (speed) of the object in feet per second,

h_o represents the initial height of the object in feet.

Activity 3

1.

Which characteristics of this situation indicate that you can model it using a quadratic
function?

2.

Write a function for the height of the pumpkin, h(t), in terms of time, t.

3.

Does the function you wrote have a minimum or maximum? How can you tell from the
form of the function?

Suppose that a catapult hurls a pumpkin from a height of 68 feet at an initial vertical
velocity of 128 feet per second.

Activity 3

4.

Use technology to graph the function. Sketch your graph and label the axes.

5.

Use technology to determine the maximum or minimum point and label it on the graph. Explain what it means in terms of the problem situation.

Activity 3

6.

Determine the y-intercept and label it on the graph. Explain what it means in terms of the problem situation.

7.

Use a horizontal line to determine when the pumpkin reaches each height after being catapulted. Label the points on the graph.

a)

128 feet

b)

260 feet

c)

55 feet

8.

Explain why the x- and y-coordinates of the points where the graph and each horizontal line intersects are
solutions.

Activity 3

9.

When does the catapulted pumpkin hit the ground? Label this point on the graph. Explain how you determined your answer.

The time when the pumpkin hits the ground is one of the x-intercepts, (x, 0). When you use an
equation to model a situation, you refer to the x-coordinate of the x-intercept as the root. The root of
an equation indicates where the graph of the equation crosses the x-axis.

The zeros of a

function are the

x-values when the

function equals 0.

REMEMBER...

Expressing a Quadratic Function as the Product of Two
Linear Functions

Now let’s consider a real-world situation that you can model with a quadratic function in a different form.

Activity 4

The Jacobson brothers own and operate their own ghost tour business. They take tour groups around town on a
bus to visit the most notorious “haunted” spots throughout the city. They charge $50 per tour. Each summer, they
book 100 tours at that price.

The brothers are considering a decrease in the price per tour because they think it will help them book more tours. They estimate that they will gain 10 tours for every $1 decrease in the price per tour.

Activity 4

1.

According to the scenario, how much money do the Jacobson brothers currently generate each summer with
their ghost tour business?

Revenue is the amount of money regularly coming into a business. In the ghost tour business, the revenue is the
number of tours multiplied by the price per tour. You can refer to your response to Question 1 as revenue. Because the Jacobson brothers are considering different numbers of tours and prices per tour, you can model the revenue using a function.

2.

Write a function, r(x), to represent the revenue for the ghost tour business.

a)

Let x represent the decrease in the price per tour. Write an expression to represent the number of tours booked if the decrease in price is x dollars per tour.

Activity 4

b)

Write an expression to represent the price per tour if the brothers decrease the price x
dollars per tour.

c)

Use your expressions from parts (a) and (b) to represent the revenue, r(x), as the number of
tours times the price per tour.

Revenue = Number of Tours •

Price per Tour

You can always check your function

by testing it with

values of x. What

is the value of r(x)

when x = 0? Does it

make sense?

THINK ABOUT...

Activity 4

3.

Use technology to graph the function r(x). Sketch your graph and label the axes.

4.

Assume that the Jacobson brothers’ estimate that for every $1 decrease in the price per tour, they will
gain 10 tours is accurate.
a)

What is the maximum revenue that the Jacobson brothers could earn for the summer?

Don’t forget to label

key points!

REMEMBER...

SUMMARY

• Graphs of quadratic functions have symmetry, intervals of increase and decrease, a limited range, and a maximum of two x -intercepts.

• The coefficient of x2 in a quadratic function determines whether the graph looks like a U or a ∩.

• You call the x -coordinate of each x -intercept of a quadratic equation a root. The roots of an equation indicate the x -values when y = 0 .

• You can write a quadratic function as the product of two linear functions.

The citizens of Herrington County have an existing dog park for dogs to play but have
decided to build another one so that one park will be for small dogs and the other will be
for large dogs. The plan is to build a rectangular fenced in area adjacent to the existing
dog park, as shown in the sketch. The county has enough money in the budget to buy
1000 feet of fencing.

1.Determine the length of the new dog park, l, in terms of the width, w.

2.Write the function A(w) to represent the area of the new dog park as a function of the width, w .

3.Does this function have a minimum or a maximum point?

PRACTICE

PAGE 671

The citizens of Herrington County have an existing dog park for dogs to play but have
decided to build another one so that one park will be for small dogs and the other will be
for large dogs. The plan is to build a rectangular fenced in area adjacent to the existing
dog park, as shown in the sketch. The county has enough money in the budget to buy
1000 feet of fencing.

4.Determine the x-intercepts of the function (write in the form of a (x, y) coordinate
point no spaces).

5.What should the dimensions of the dog park be to maximize the area? What is the maximum area of the park?

Width =

Length =

Area =

PRACTICE

PAGE 671

Video Response

Share ONE thing you learned about quadratic functions OR ONE thing you still have questions about.

(This is a video response, only the teacher will see it.)

Open Video Recorder

After Class Assignment

• Make sure to submit OPT 4: Day 02A WORKBOOK
  Assignment M5.T1.L1 Exploring Quadratic Functions

• Spend at least 15 minutes reviewing your notes.

• Write down any questions you have.

• Make a plan to get help.