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 Depdendent Variable

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 Dependent Variable

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# Parabolas Tutorial

Section 6.1: Parabolas

Learning Objectives:
1. Graph parabolas in which the vertex is the
origin.
2. Find the equation of a parabola.
3. Graph parabolas in which the vertex is not
the origin.
4. Solve applied problems involving
parabolas.
1. The Parabola

Parabolas that Open Up or Down
The graph of y = a(x – h)2 + k or y = ax2 + bx + c is a parabola that
1. opens up if a > 0 and opens down if a < 0.
2. has vertex (h, k) if the equation is of the form y = a(x – h)2+ k.
3. has a vertex whose coordinate is The y-coordinate
is found by evaluating the equation at the x-coordinate of the
vertex

y = a(x – h)2 + k, a > 0

y = a(x – h)2 + k, a < 0

1. The Parabola

A parabola is the collection of all points P in the plane that are
the same distance from a fixed point F as they are from a fixed
line D. The point F is called the focus of the parabola, and the
line D is its directrix. In other words, a parabola is a set of points
2. Equations of a Parabola

From the distance formula we can obtain the fact that the equation
of the parabola whose vertex is at the origin and opens to the right is
y2 = 4ax.

 Equations of a Parabola: Vertex at (0, 0); Focus on an Axis; a > 0 Vertex Focus Directrix Equation Description (0, 0) (a, 0) x = – a y2 = 4ax Parabola, axis of symmetry is the x-axis; opens to the right (0, 0) (– a , 0) x = a y2 = – 4ax Parabola, axis of symmetry is the x-axis; opens to the left (0, 0) (0, a) y = – a x2 = 4ay Parabola, axis of symmetry is the y-axis; opens up (0, 0) (0, – a) y = a x2 = – 4ay Parabola, axis of symmetry is the y-axis; opens down
Graphing a Parabola

Example continued: Graph the equation x2 = – 16y.
Finding the Equation of a Parabola

Example: Find an equation of the parabola with vertex at (0, 0)
and focus at (– 8, 0).
3. A Parabola Whose Vertex is Not the Origin

 Parabolas with Vertex at (h, k); Axis of Symmetry Parallel to a Coordinate Axis, a > 0 Vertex Focus Directrix Equation Description (h, k) (h + a, k) x = h – a (y – k)2 = 4a(x – h) Parabola, axis of symmetry parallel to x-axis; opens to the right (h, k) (h – a, k) x = h + a (y – k)2 = – 4a(x – h) Parabola, axis of symmetry parallel to x-axis; opens to the left (h, k) (h, k + a) y = k – a (x – h)2 = 4a(y – k) Parabola, axis of symmetry parallel to y-axis; opens up (h, k) (h, k – a y = k + a (x – h)2 = – 4a(y – k) Parabola, axis of symmetry parallel to y-axis; opens down
A Parabola Whose Vertex is Not the Origin

Example continued: Graph the parabola x2 + 2x – 8y + 25 = 0.