Hyperbolas Ellipses Inside Out (9.4)

POD Sketch the ellipse. What is the center? What are the lengths of the three radii?

POD Sketch the ellipse. What is the center? (2,-3)

POD Sketch the ellipse. What are the lengths of the three radii? x-radius = 4, y-radius = 5, focal radius = 3 What does it mean that the y-radius is longer? So, what are vertices and what are their coordinates?

POD Sketch the ellipse. What does it mean that a is the y-radius? The ellipse is vertical. The vertices are located at (2, 2) and (2, -8). How do you determine the foci? What are they?

POD Sketch the ellipse. Center at (2,-3) x-radius = 4 y-radius = 5 focal radius = 3 foci (2, 0) and (2,-6) vertices (2, 2) and (2, -8)

New conic form Consider the original form of the POD equation. What is the same between the coefficients for x 2 and y 2 ? What would change in the graph if we changed the sign of one of them?

New conic form Let’s look at an equation in which those coefficients have different signs.

New conic form Let’s look at an equation in which those coefficients have different signs. The algebra is the same. Be careful with negative signs and parenthesis!

New conic form Let’s look at an equation in which those coefficients have different signs. The algebra is the same. Be careful with negative signs and parenthesis! What do you think the graph will look like?

New conic form This is an equation for a hyperbola. Like an ellipse, it has a center, and three radii. Instead of circling in around the center, it opens out from the center. What do you think the center is? Could you make a guess about the x-radius and y-radius? Or even the focal radius?

Graphing hyperbolas This process is a little more complicated than the one for graphing ellipses. It involves using a rectangle.` Let’s draw that rectangle, once we’ve rewritten the equation. 1.Mark the center. 2.Mark out the x-radius in both directions. 3.Mark out the y-radius in both directions. 4.Draw a rectangle along both of those lengths.

Graphing hyperbolas 1.Mark the center. (2,-3) 2.Mark out the x-radius in both directions. Length = 4 3.Mark out the y-radius in both directions. Length = 5 4.Draw a rectangle along both of those lengths.

Graphing hyperbolas With the rectangle in place: 1.Draw diagonals from the center to the corners. 2.The hyperbola opens out along the axis with the positive sign, in this case along the x-axis. Draw the curves at the rectangle, so the diagonals are asymptotes.

Graphing hyperbolas With the hyperbola in place: 1.Find the vertices. 2.Find the foci– they’re tucked inside the curves. The long radius is still a, and the short radius is still b. The equation is now c 2 = a 2 + b 2 and we can use it to find the focal radius, c.

Graphing hyperbolas x-radius = a = 4 y-radius = b = 5 Opens along x-radius Center (2,-3) Vertices (-2, -3), (6, -3) Foci (2-√41, -3) (2+√41, -3)