Introduction to Hyperbolas

Conic Sections

Hyperbola​

By Dr KEE CH

Learning objectives:

1. Students will be able to differentiate the four conic sections.

2. Students will be able to conceptualize the key features of a hyperbola.

​3. Students will be able to use an online graphing tool to graph a hyperbola.

Revision : Parabola

Multiple Choice

What is the vertex of the parabola: y=2(x-3)²+4

• (3,4)
• (-3,4)
• (3, -4)
• (-3,-4)

Multiple Choice

The focus is always inside the parabola.

• true
• false

Multiple Choice

All parabolas with a vertical directrix open the left or right.

• true
• false

Multiple Choice

True or False?
When the x-part is squared, the parabola opens up or down.

• True
• False

Multiple Choice

What direction does the parabola point?

• Up 
• Down
• Left 
• Right

Multiple Choice

(y-4)² = -8(x+1)
What are the coordinates for the focus?

• (1,4)
• (-1,6)
• (-1,2)
• (-3,4)

Multiple Choice

(x+3)² = 4(y+5)
What is the equation of the directrix?

• x = 4
• x = 2
• y = -4
• y = -6

Revision : Circle

Multiple Choice

Find the graph to this equation $$\left(x+9\right)^2+\left(y+4\right)^2=48$$  

Multiple Choice

Find the graph to this equation $$\left(x-3\right)^2+y^2=4$$  

Multiple Choice

Find the graph to this equation $$\left(x-7\right)^2+\left(y+1\right)^2=1$$  

Multiple Choice

Find the standard equation of this graph

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

• $$\left(x+1\right)^2+\left(y-4\right)^2=16$$

• $$\left(x-1\right)^2+\left(y+4\right)^2=16$$

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

Multiple Choice

Find the standard equation of this graph

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

• $$\left(x+3\right)^2+\left(y+1\right)^2=4$$

• $$\left(x-3\right)^2+\left(y-1\right)^2=4$$

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

Revision : Circle

Multiple Choice

What are the vertices of the ellipse?

• (0, 0)
• (7, 1) and (-1, 1)
• (3, -5) and (3, 7)
• (3, 1)

Multiple Choice

What is the center of the shifted ellipse?

• (0,0)
• (-3,1)
• (1,-3)
• (3,-1)

Multiple Choice

Find the standard equation of the given ellipse

Multiple Choice

Find the standard equation of the given ellipse

Multiple Choice

Which of the following is the graph of  $$\frac{\left(x+3\right)^2}{4}+\frac{\left(y+1\right)^2}{9}=1$$  

Hyperbola

Hyperbola

• Has two equal parts, same size and shape

• Symmetrical

• Approach the axes, but never touche them

• No value of x when y=0, and for y when x=0 (no x-intecepts and no x-intercepts)

• A line that a graph approaches but never intersect is called asymptote

Hyperbola y=2/x

• Asymptote: x=0 and y=0

Plot the graph ​using Desmos / Geogebra.

What did you noticed?​

• Horizontal asymptote: y=0

• Vertical asymptote: x=1

• y-intercept = -2

Check your understanding

Multiple Choice

Which type of conic is:

• Circle

• Ellipse

• Hyperbola

• Parabola

Multiple Choice

Which of the followings is true?

• y-intercept =-1

• y-intercept = 1

• y-intercept = -3

• y-intercept =3

Multiple Choice

Find the equation of hyperbola.

• $$\frac{\left(x-2\right)^2}{16}-\frac{\left(y+3\right)^2}{20}=1$$

• $$\frac{\left(x+3\right)^2}{16}-\frac{\left(y-2\right)^2}{20}=1$$

• $$\frac{\left(x-2\right)^2}{20}-\frac{\left(y+3\right)^2}{16}=1$$

• $$\frac{\left(y+3\right)^2}{16}-\frac{\left(x-2\right)^2}{20}=1$$

Multiple Choice

What are the asymptotes of the hyperbola in the picture?

• y = x and y = –x

• y = 2x and y = –2x

• y = ¹/₂x and y = –¹/₂x

• y = 2x and y = 4x

Multiple Choice

What is the center of this hyperbola?

• Center $$\left(-1,\ 4\right)$$

• Center $$\left(1,\ 4\right)$$

• Center $$\left(-1,\ -4\right)$$

• Center $$\left(1,\ -4\right)$$

Open Ended

What is your key takeaway from today's lesson?