Recap on Parabola, Vertex and Axis of Symmetry

Curved Graphs

Plotting Parabola

Objectives

• Construct a table of values for quadratic

• Plot the graph of a parabola from a table of values

What is a parabola?

• A parabola is a smooth U-shaped curve. 

• An equation of the form y = ax² + bx + c will graph to give a parabola.

• This lesson concentrates on parabolas of the form y = ax² + c.

Read the equation of a parabola

• You can draw a quick sketch of a parabola using the information from its equation, y = ax² + c.

• The coefficient of x², a, tells you the DIRECTION and WIDTH of the parabola.

Read the equation of a parabola

• You can draw a quick sketch of a parabola using the information from its equation, y = ax² + c.

• The constant value, c, in the equation of a parabola, tells you the y-intercept. Changing the constant term c moves the curve up or down the axis.

The axis of symmetry and vertex

• The axis of symmetry is the line along which the graph can be folded so that both halves of the parabola overlap.

• The vertex is the point at which the parabola turns.

• For y = ax² + c, the axis of symmetry is the y-axis (x = 0)

• For y = ax² + c, the vertex is the y-intercept (0,c).

The domain and range

• The domain represents the possible x values of the parabola and the range gives the possible y values

• For the parabola y = x² + 2:

• the domain is: all real values of x

• the range is: y ≥ 2

Describing a parabola (1)

• Concave up

• x-intercepts = –8 and 8 y-intercept = –1

• Minimum = –1

• Domain: all real x Range: y ≥ –1

• Vertex: (0, –1)

Describing a parabola (2)

• Concave down

• x-intercepts = –5 and 5 y-intercept = 7

• Maximum = 7

• Domain: all real x Range: y ≤ 7

• Vertex: (0, 7)

Describing a parabola (3)

• Concave up

• No x-intercepts , y-intercept = 2

• Minimum = 2

• Domain: all real x Range: y ≥ 2

• Vertex: (0, 2)

Describing a parabola (4)

• Concave down

• x-intercepts s = - V5 and V5, y-intercept = 5

• Maximum = 5

• Domain: all real x Range: y ≤ 5

• Vertex: (0, 5)

Multiple Choice

If graph f is y = x², what will the graph of g (red curve) be?

• y = 3x²

• y= - 3x²

• y = 1/3 x²

Multiple Choice

Which of the following is true about the graph?

• maximum = -4

• minimum = -4

Multiple Choice

Which of the following is true about the graph?

• y-intercept = -4

• y-intercept = -2 and 2

Multiple Choice

The graph of a quadratic function is called

• a line
• a hyperbola
• a parabola
• a cubic

Multiple Choice

Standard form of a quadratic equation

• y=x²
• y=ax²+bx+c
• y=mx+b
• y=x

Multiple Choice

3x² - 4 is a quadratic

• true
• false

Multiple Choice

Which of the following is NOT a quadratic function

• f(x) = -3x²
• f(x) = -5x - x²
• y = 3x³ - 4x + 5
• A = pi.r²

Multiple Choice

What are the green dots called?

• axis of symmetry
• vertex
• parabola
• roots or x-intercepts

Multiple Choice

Does this parabola have a maximum or minimum?

• Maximum
• Minimum

Multiple Choice

What are the x- intercepts?

• x= 0 and x= -4
• x= 0 and x= 4
• y= 0
• x= 2

Multiple Choice

The formula x = -b/2a is used to find

• the roots
• the x-value of the vertex
• the y-intercept
• the axis of symmetry

Multiple Choice

What is the vertex of the quadratic function:
y = x² - 6x + 11?

• (3,2)
• (-3,-2)
• (-3,2)
• (3,-2)

Multiple Choice

When a parabola opens downward the vertex is

• a minimum
• a maximum
• a vertex
• a parabola

Multiple Choice

Does the equation Up or or down?

Y = -3x² +7x - 2

• up

• down

Multiple Choice

Does this question have a maximum or a minimum?

• Maximum
• Minimum

Multiple Choice

What is the equation for axis of symmetry?

• y=3
• x=3
• x=5
• x=1

Multiple Choice

The axis of symmetry line passes through which point?

• y-intercept
• any random point
• vertex
• none of these

Multiple Choice

Which equation is in standard form?

• y=a(x-h)²+k

• y=ax²+bx+c

Multiple Choice

Where is the axis of symmetry for this parabola?

• x = −1
• x = 0
• y = 3
• x = −3

Multiple Choice

$$f\left(x\right)=ax^2+bx+c$$  

• Quadratic Function

• Linear Function

• Polynomial

• Trinomial

Multiple Choice

Where the minimum or maximum value of a quadratic function occurs

• Vertex of a Parabola

• Axis of Symentry

• Parent Function

• Quadratic

Quadratic Functions

● Using Vertex Form

● Using Standard Form

Quadratic Functions

● Using Vertex Form

● Using Standard Form

Quadratic Functions

● Using Vertex Form

● Using Standard Form

Multiple Choice

Write the vertex form of the equation that has the indicated vertex and passes through the given point.
Vertex (-1,4); point (1,0)

$$f\left(x\right)=a\left(x-h\right)^2+k$$  

• $$f\left(x\right)=-\left(x+1\right)^2+4$$  

• $$f\left(x\right)=\left(x+1\right)^2+4$$  

• $$f\left(x\right)=-\left(x+4\right)^2+1$$  

• $$f\left(x\right)=-\left(x+1\right)^2$$  

Multiple Choice

Write the vertex form of the equation that has the indicated vertex and passes through the given point.
Vertex (-2,-1); point (0,3)

$$f\left(x\right)=a\left(x-h\right)^2+k$$  

• $$f\left(x\right)=-\left(x+1\right)^2+3$$  

• $$f\left(x\right)=\left(x+2\right)^2-1$$  

• $$f\left(x\right)=-\left(x+5\right)^2+7$$  

• $$f\left(x\right)=\left(x-3\right)^2$$  

Multiple Choice

Write the vertex form of the equation that has the indicated vertex and passes through the given point.
Vertex (-2,5); point (0,9)

$$f\left(x\right)=a\left(x-h\right)^2+k$$  

• $$f\left(x\right)=\left(x+5\right)^2+2$$  

• $$f\left(x\right)=\left(x-5\right)^2$$  

• $$f\left(x\right)=\left(x+2\right)^2-5$$  

• $$f\left(x\right)=\left(x+2\right)^2+5$$  

Multiple Choice

If given the equation y = 3(x + 5)² - 4, what is the vertex of the parabola?

• (5,-4)

• (-5,-4)

• (-15, 4)

• (15, -4)

Vertex Form of a Quadratic Function

​

Learning Objectives

• Determine the vertex of a quadratic function

• Graph a quadratic function in vertex form

Multiple Select

Check all of the equations that are in standard form

• $y=2x+1$

• $y=x^2+4x-9$

• $y=4\left(x-5\right)^2+3$

• $y=x^2+9$

Multiple Select

Select all equations that are in vertex form

• $y=x^2+5x$

• $y=3\left(x+4\right)^2-1$

• $y=-\left(x+3\right)^2-2$

• $y=\left(x+3\right)\left(x-2\right)$

Multiple Choice

If given the equation y = 3(x + 5)² - 4, what is the vertex of the parabola?

• (5,-4)

• (-5,-4)

• (-15, 4)

• (15, -4)

Steps to graphing a quadratic equation in Vertex Form

• Identify the vertex (h,k)

• Select 2 more x's close to one side of the vertex to plug into the equation.

• Reflect those two points to the other side of the axis of symmetry

Poll

Did you read the previous slide?

• Yes

• No

Multiple Choice

You already have that the equation y = 3(x + 5)² - 4 has a vertex at (-5, -4). Click on the image to enlarge. The next step is to pick 2 x's close to the axis of symmetry and on the same side of the axis of symmetry. What would you pick

• -6 and -7

• -6 and -4

• -5 and -6

• -5 and -7

Poll

If you plug in -6 and -7 for x you'll get the two points which are now graphed. When a graph has an axis of symmetry, it means the graph is same on both sides. Do you see that you could reflect the new points across the axis of symmetry to get two more points on the other side? Click on image to make it larger.

• Yes

• No

Open Ended

Here's what your graph should look like. What questions do you have? Click on the image to view the graph.

Multiple Choice

Sketch the graph of the following function

 $y\ =\ 2\left(x+4\right)^2-4$  

Multiple Choice

Sketch the graph of the following function

 $y\ =\ -\left(x-3\right)^2-3$