3/20 - Characteristics of graphs

​Remove Your
Hoodie

Put up ALL
headphones/ earbuds

Prepare to Relinquish your cellphone until the end of class.

Do Now

7

Solve each equation for y.

1.

2y = x + 4

2. 8x + 4y = 16

LO: SWBAT identify key features in mathematical and real-world problems of linear, exponential, and quadratic functions.

DOL: Given 5 problems, I will be able to identify key features in mathematical and real-world problems of linear, exponential and quadratic functions of at least 4 of 5 questions.

8

Today you will graph and identify key

feature of …

linear, quadratic, and exponential

functions.

12

Vocabulary

9

y-intercept

x-intercept

Δy = 4

Δx = 6

Slope = — = — = —
Δy

6

Δx

4

3
2

● Slope-Intercept Form:

y = mx + b

● Point-Slope Form:

y − y1 = m(x − x1)

● Standard Form:

Ax + By = C

Linear function

Vocabulary

10

Vertex
(max/min)

x-intercepts

Parabola

y-intercept

Axis of

symmetry

x-intercepts (or zeros)
represent the solutions
of a quadratic equation,
or f(x) = 0.

● Standard Form:

y = ax² + bx + c

● Vertex Form:

y = a(x − h)² + k

Quadratic function

Vocabulary

11

Exponential function

Form:y = abx

Horizontal asymptote

y-intercept

Exponential growth
-Nonlinear increase
of form bx (b > 1)

Exponential decay
-Nonlinear decrease
of form bx (0 < b < 1)

Multiple Select

What characteristics do linear, quadratic and exponential functions have in common.

Select all that apply.

• (possible)

  y-intercept

• (possible)

  x-intercept

• axis of symmetry

• horizontal asymptote

• Vertex

​We will start with
Linear Functions

Vocabulary

9

y-intercept

x-intercept

Δy = 4

Δx = 6

Slope = — = — = —
Δy

6

Δx

4

3
2

● Slope-Intercept Form:

y = mx + b

● Point-Slope Form:

y − y1 = m(x − x1)

● Standard Form:

Ax + By = C

Linear function

To graph a Linear Function on Desmos
it must be in Slope Intercept Form

Slope Intercept Form

y = mx + b

Standard Form

Ax + By = C

Example: What is the
y-intercept of the graph of
y = 11x – 23?

Example: What is the
y-intercept of the graph of
6x – 12y = 24?

y = 11x – 23

This one requires a

little more work!!!

Standard Form to Slope Intercept Form

Standard Form

Ax + By = C

Example: What is the slope of the graph of 6x – 12y = 24?

Option 1: Put the equation in slope intercept form (solve for y). 3 min

14

Multiple Choice

Put the equation in slope-intercept form:
2x - 4y = 8

• y = 1/2x - 2
• y = 2x + 2
• y = 4x + 2

Multiple Choice

Which equation in standard form represents this graph?

• -3x +2y = -6

• 3x + 2y = 6

• -3x - 2y= 12

• -3x + 2y = -12

Vocabulary

10

Vertex
(max/min)

x-intercepts

Parabola

y-intercept

Axis of

symmetry

x-intercepts (or zeros)
represent the solutions
of a quadratic equation,
or f(x) = 0.

● Standard Form:

y = ax² + bx + c

● Vertex Form:

y = a(x − h)² + k

Quadratic function

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y - intercept:

x - intercept:
(zeros, roots, solutions)

Vertex:
(written as a point (x,y))

Axis of symmetry:
(written as an equation; x = )

Maximum or
Minimum:

(0, -4)
(Opens down )

(Opens up )

Identifying Key Attributes

Labelling

Identify key attributes of the quadratic

Drag labels to their correct position on the image

x = 0

y = 4

Min

(-2, 0)

(0, 4)₂

(0, 4) ₁

Max

(2, 0)

Labelling

Identify key attributes of the quadratic

Drag labels to their correct position on the image

(2, 4)

x = 2

y = 4

(4, 0)

(0, 0) ₁

(0, 0)₂

Min

Max

Multiple Choice

Which graph has a minimum of (-3, -10)?

• A

• B

• C

• D

Recall: Exponential Functions

21

Exponential function

Form y = abx

Horizontal asymptote

y-intercept

Exponential growth
-Nonlinear increase
of form bx (b > 1)

Exponential decay
-Nonlinear decrease
of form bx (0 < b < 1)

22

y-Intercepts of Exponential Functions

Consider the general exponential function y = ab^x.

The y-intercept is defined as the y-value where x = 0.

If x = 0, then y = ab^x = ab⁽⁰⁾

← Remember the zero power rule? (x0 = 1)

a(1)

a

← The y-intercept is the a-value

For exponential functions of form y = abx, the y-intercept is (0,a).

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What about b?

So the a in abx represents the y-intercept. What about the b?

There are two cases:

0 < b < 1

What is the difference between

these two cases?

For smaller values of b,
this will cause the output
to decrease.

For larger values of b, this
will cause the output to
increase.

In exponential graphs this
is called growth.

In exponential graphs this
is called decay.

Match

Match the following

• a

• $$y\ =\ ab^x$$

• b

Growth

Decay

y-intercept

Exponential Function Equation

Growth/Decay Factor

• Growth

• Decay

• y-intercept

• Exponential Function Equation

• Growth/Decay Factor

Math Response

What is the "a" or "y-intercept" of the given

equation of an exponential function?

$$y\ =\ 5\left(\frac{1}{2}\right)^x$$

Type answer here

Correct Answer: 5

Deg^°

Rad

Multiple Choice

Describe the function. $$y\ =\ 2\left(\frac{1}{2}\right)^x$$

• Exponential Growth

• Exponential Decay

Multiple Choice

Which of the below exponential graphs shows decay?

Demonstration of Learning

DOL: Given 5
problems, students
will correctly graph
linear, quadratic,
and exponential
functions on the
coordinate plane
and identify key
features in
mathematical and
real-world
problems in at
least 4 of 5
questions.

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Open Ended

• What characteristics do linear, quadratic and exponential functions have alike and different?

Use complete sentences.