Key Features LESSON, Function Unit

Key Features LESSON, Function Unit

• ​Domain and Range

• ​x-intercepts and y-intercepts

• ​Behavior of graph, constant, decreasing, increasing

• ​Continuous or discontinuous

• ​Maxima or minima points

• ​Specialty features for graphs such as: vertex, axis of symmetry, asymptotes

• ​.....and many more....

​Recap from previous lesson (only write if you don't know...)

​x-intercept: point or points a graph intersects the x-axis

​

y-intercept: point or points a graph intersects the y-axis

​

domain: x-values for a graph; can be discrete or continuous

​

​range: y-values for a graph; can be discrete or continuous

​

​maximum point: the highest point on a graph or highest points if referring to interval

​

​minimum point: the lowest point on a graph or lowest points if referring to interval

​

Multiple Choice

What is the range of the graph?
Remember: Range is y! 

• 1 ≤ x ≤ 5
• 1 < x < 5
• 1 ≤ y ≤ 5
• 1 < y < 5

Multiple Choice

What is the domain of the graph?
Remember:   Domain is the "x" values from left to right

• -7 ≤ x < 5
• -3 ≤ x < 1
• -3 ≤ y < 1
• -7 ≤ y < 5

Multiple Choice

What is the RANGE?

• y < 4

• y ≥ 4

• y ≤ 4

• All Real Numbers

Multiple Choice

What is the domain of the graph?

• x ≥ -6

• -7 ≤ x ≤ 3

• -8 ≤ x ≤ 4

• x ≤ -6

• All real numbers

Multiple Choice

What is the range of the graph?

• y ≥ 1

• 1 ≤ y ≤ 5

• 1 ≤ y ≤ 4

• y ≤ 5

• All real numbers

Multiple Choice

What is the domain of the graph?

• (-∞,∞)
• [-1,∞)
• (-1,∞)
• [-10,10]

Multiple Choice

What is the domain of the following graph?

• (-∞,∞)
• {1.5}
• [-5,5]
• (-5,5)

Multiple Choice

What is the range of the graph?

• (-∞,∞)
• [-2,∞)
• [0,∞)
• [3,∞)

Multiple Choice

What is the domain of the graph?

• [-2,2]
• [-2,1]
• [-4,-1]
• (-∞,∞)

Multiple Choice

What is the y-intercept shown on the graph?

• (0, -2)
• (4, -2)
• (4, 0)
• (0, 4)

​Key Features for a parabola:Draw and label in your notes

​Vertex Point will always be either the MAXIMUM or MINIMUM point of a parabola, this is a minimum point because it is the lowest point

​Reminder:

​Quadratic is a polynomial with a degree (highest exponent) of 2

​The graph of the quadratic is called a PARABOLA.

Multiple Choice

What is the green dashed line called?

• x-intercepts

• parabola

• axis of symmetry

• line of dashes

Multiple Choice

What is the green dot on the parabola called?

• vertex

• y-intercept

• x-intercept

• axis of symmetry

Multiple Choice

Identify the vertex.

• (4, -4)

• (2, 0)

• x² – 8x + 12

• (0, 12)

Multiple Choice

Identify the vertex.

• (0, 3)

• (-2, -1)

• (-3, 0)

Multiple Choice

What is the green dot on the parabola called?

• maximum
• minumum
• roots
• zeros

Multiple Choice

Does this graph have a maximum or minimum?

• Minimum
• Maximum

Multiple Choice

Is this a max or min?

• max
• min

Multiple Choice

What is the green dashed line called?

• roots or x-intercepts
• parabola
• axis of symmetry
• line of dashes

Multiple Choice

Which best describes the vertex of a parabola?

• Any point on the parabola.
• A point on the x-axis.
• The highest or lowest point of a parabola
• A point on the y-axis.

Multiple Choice

The axis of symmetry is a line that goes through the...

• y-intercept
• any random point
• vertex
•  x-axis or y-axis

​x-intercepts will always have zero for the y-coordinate

​ ​Examples of x-intercepts would be:

​ (-3, 0) (4, 0) (10, 0) (-0.75, 0)

​

​y-intercepts will always have zero for x-coordinate

​ Examples would y-intercepts would be

​ (0, 10). (0, -4). (0, 1/2)

​

​When determining x or y intercepts from equation, use this rule to substitute zero in for y to find the x-intercepts and then zero in for x to find the y-intercepts

​Example: 4x - y = 12 x-intercept: 4x - 0 = 12 y-intercept: 4(0) - y = 12 4x = 12 -y = 12

​ x = 3 or (3, 0) y = -12 or (0, 12)

​You could always GRAPH the equation and look at the graph to determine x and y intercepts

Multiple Choice

Find the axis of symmetry:

$$f\left(x\right)=6x^2-2$$

• x = 6

• x = 0

• x = 2

• x = -2

Multiple Choice

Given the list of ordered pairs, what is the x-intercept?
(8,10),  (3, -4),  (0, 8), (4, -3), (9, 0)

• (8,10)
• 8, 3, 0, 4, 9
• (0,8)
• (9,0)

Multiple Choice

What is the y-intercept?

• (-12, 0)

• (0, -12)

• (12, 0)

• (0, 12)

Multiple Choice

Which word is NOT a synonym for x-intercept?

• zero
• y-intercept
• solution
• root

To calculate the vertex point:​

Multiple Choice

What's the axis of symmetry of y = 3x² - 6x + 4

• x=1
• x=-6
• x=2
• x=-1

Multiple Choice

Find the vertex of 
f(x) = -x² - 4x + 12

• (-2, 16)
• (2, 0)
• (2, 4)
• (-2, 4)

Multiple Choice

What is the x-intercept?

• 12
• -3
• 8
• 0

Multiple Choice

What is the y-intercept?

• 12

• -3

• 8

• 0

Multiple Choice

Does this graph have a minimum or maximum? What point?

• minimum at (-3, -5)

• minimum at (3, 5)

• maximum at (-3, -5)

• maximum at (3, 5)

Multiple Choice

Does this graph have a minimum or maximum? What point?

• Minimum at (0, 0)

• Minimum at (0, 2)

• Maximum at (0, 0)

• Maximum at (0, 2)

​Reminder:

​x-intercepts of a polynomial are also referred to as:

​ zeros

​ roots

​ solutions

​

Multiple Choice

What are the x and y intercepts of the graph?

• x-intercepts = -4, -1, 2, 1 multiplicity 2

  y-intercept = -8

• x-intercepts = -4, -1, 1, 2

  y-intercept = 0

• y-intercepts = -4, -1, 2, 1 multiplicity 2

  x-intercept = -8

• x-intercepts = 4, 1, -2, -1 multiplicity 2

  y-intercept = -8

Multiple Choice

What are the zeros and y-intercept of the function?

• x-intercepts = 4,1

  y-intercept = 3

• x-intercepts = 3,1,-2,-5

  y-intercept = 3.5

• x-intercepts = -3, -1, 2, 5

  y-intercept = 3

• x-intercepts = -2,4

  x-intercepts = -7

Multiple Choice

What are the x and y intercepts of the graph? 

• x-intercepts = -2, 1, 3
  y-intercept = 3

• x-intercepts = 2, -1, -3
  y-intercept = 3

• x-intercepts = -2, 1, 3
  y-intercept = -2

• x-intercepts = -2, 1, 3
  y-intercept = -2

​Asymptotes

​Asymptotes are (imaginary lines) that a graph never crosses

​Consider this function: f(x) = 2^x

​The graph has an asymptote at y=0 because, the graph--IF drawn forever--would never touch the x-axis which is the line y=0.

​Logarithmic Function

​Has an asymptote at x = 0 also known as the y-axis because if you could drawn the graph FOREVER, the graph would NEVER cross the y-axis.

​

​

Multiple Choice

What are the asymptotes?

• x=-2 and y=-1
• x=-2 and y=1
• x=2 and y=1
• x=2 and y = -1

Multiple Choice

What is the equation of the horizontal asymptote of the given function?

• y = 2
• x = 1
• There isn't one
• y = 5

Multiple Choice

What is the horizontal asymptote?

• y = 2
• y = -2
• x = 2
• y = 0

Multiple Choice

What are the equations of the asymptotes of the given graph

• x=1, x= 2, y =1, y= 2
• x= 2, x= -2, y = 1
• x=2 and  y= -1
• x=1, y =2, y = -2

​You can also determine asymptotes by looking at the equation

​You can NOT have zero in the denominator and this creates VERTICAL asymptotes:

​

​x can not be 0 or -3

​There would be vertical asymptotes at 0 and -3

​x can not be -1, 0

​There would be vertical asymptotes -1 and 0

Multiple Choice

What are the Vertical Asymptotes of...

• x = 0, -5
• x=0, 5
• x = 2, -7
• x = -2, 7

Multiple Choice

What is the Vertical Asymptotes? 

• x= -5
• x= 5
• x= 6
• x= -6

Multiple Choice

State the vertical asymptote.

• x = -2/5

• x = -5/2

• x = -5

• x = 2