Alg1 Lesson 3.5: Graphs and the Factored Form of a Quadratic

Lesson 3.5: Graphs and the
Factored Form of a Quadratic

Obj: 6B, 6C, 7B, (7C), 10D, 10E, 10F: I can determine an
algebraic rule for a quadratic function given a sufficient
number of points from the graph.

EQ: How do I formulate a quadratic equation given three
points?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Poll

What is the most important role?

• Scribe

• Facilitator

• Resourcer

• Includer

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Warming Up with

Graphs and Factors

Multiple Choice

If a quadratic function has a factor of (x − 3), what would be an x-intercept for the graph of
the function?

• (3, 0)
• (-3, 0)

• (0, -3)

• (0, 3)

Multiple Choice

If a quadratic function has a factor of (x − 3) and you evaluate the function at x = 3, what happens?

• The value of the function is 0.
• The value of the function is 3.
• The value of the function is -3.

• The value of the function is -6.

Multiple Choice

Does it always happen that when you substitute an x-coordinate of an x-intercept into a quadratic function, it makes the value of one of the factors zero?

• Yes, substituting an x-coordinate of an x-intercept makes one of the factors zero.
• No, the x-coordinate does not affect the factors.
• Sometimes yes, but only for certain quadratic functions.

• No, it can result in a non-zero value.

Multiple Choice

Suppose a quadratic function has a factor of (2x −1). What x-value will make the value of the factor zero?

• -1
• 1/2
• 0
• 1
• -1/2

Part 2: Determining the

Scale Factor

Draw

Draw two different parabolas (in different colors) that will have x-intercepts at (−4, 0) and (1, 0).

Multiple Choice

How many parabolas do you think you could draw that will have x-intercepts at (−4, 0) and (1, 0)?

• 2

• Infinitely many

• 20

• 1

Compare and Contrast these graphs.

Multiple Choice

Which graph represents f(x) = (x+4)(x-1)?

• Graph A

• Graph B

Dropdown

Can we use any points other than the x-intercepts to find out which graph represents the function? What other points do we know for sure?

We can use the

​

Select answer

-intercepts: (​

Select answer

,​

Select answer

) and (​

Select answer

,​

Select answer

).

Multiple Choice

What is the y-intercept of Graph B?

• (0, -8)
• (0, -4)

• (1, 0)

• (-4, 0)

Multiple Select

How is the y-intercept of Graph B different from the y-intercept of Graph A?

• It is 4 unit further up than (0, -4).

• It is 4 units further down than (0,−4).

• It is half as far away from the origin as (0, -4).

• It is
  twice as far away from the origin as (0,−4).

Multiple Choice

Suppose we wanted to translate the y-intercept down 4 units. What operation do you think
we could do that would move the parabola down?

• We could subtract 4 from the function f.

• We could add 4 to the function f.

• We could divide the function f by 4.

• We could multiply the function f by 4.

Multiple Choice

What is the new function rule when shifted down 4 units? Let’s call it h.

• h(x)=(x+4)(x −1)-4

• h(x)=(x+4)(x −1)+4

• h(x)=4(x+4)(x −1)

• h(x)=(1/4)(x+4)(x −1)

Multiple Choice

Suppose we wanted the y-intercept to be twice as far away. What operation could we try to
make something twice as much? We could multiply by 2.

• No, it will also have different x-intercepts.

• No, some points will be different, but it will have the same x-intercepts.

• Yes, only the y-intercept changes.

Multiple Choice

Suppose we wanted the y-intercept to be twice as far away. What operation could we try to
make something twice as much?

• We could subtract 2 from the function f.

• We could add 2 to the function f.

• We could divide the function f by 2.

• We could multiply the function f by 2.

Multiple Choice

What is the new function rule when we vertically stretch by a factor of 2? Let’s call it g.

• g(x)=(x+4)(x −1)-2

• g(x)=(x+4)(x −1)+2

• g(x)=2(x+4)(x −1)

• g(x)=(1/2)(x+4)(x −1)

Multiple Choice

Try to multiply the factored form by 2. What happens to the graph of the function? Does the y-intercept change? Do the x-intercepts change?

• The y-intercept changes but the x-intercepts stayed the same.

• The y-intercept changes and the x-intercepts change.

• None of the intercepts change.

• The y-intercept stayed the same but the x-intercepts changed.

Multiple Choice

Why do you think the x-intercepts stayed the same?

• The x-intercepts are irrelevant to the function's behavior.
• The x-intercepts are determined by the vertex of the parabola.
• The x-intercepts changed because the function was modified.
• The x-intercepts stayed the same because the points (-4, 0) and (1, 0) still yield zero when substituted into the function.

Quadratic Factored Form

A quadratic function written in factored form
is f(x)= a(x − r )(x − s), where a is not zero
and r and s are real numbers.

This graph has the same
x-intercepts as graphs A and B, but
a different y-intercept.

What is the factored form of this
graph? In other words, if f(x) =
a(x+4)(x-1) what is the value of a
for Graph C?

a can be
determined
using any 3
points on the
quadratic (not
just the
intercepts).

Part 3: The Scale Factor and

Matching Graphs and

Functions

Complete Handout 3.5

Find the x and y intercepts of each function, then match each graph to a function.

To get x-intercepts, set
the factors equal to 0 and
solve for the x’s.

To get y-intercepts,
evaluate the function
when x = 0.

Part 4: Characteristics of

Quadratic Functions

Domain and Range of Quadratics Review

Vertex Form:
y = (x - 2)² - 1

Domain:
All Real Numbers

Range: y ≥ -1

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.