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9.1 Parabolas Parabolas Parabola: the set of points in a plane that are the same distance from a given point called the focus and a given line called the directrix. Notice that the axis of symmetry is the line through the focus, perpendicular to the directrix d2 d1Focus d2 d3 d1 Vertex d3 Directrix Notice that the vertex is located at the midpoint between the focus and the directrix... Also, notice that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix... Parabola with measures.gsp We can determine the coordinates of the focus, and the equation of the directrix, given the equation of the parabola.... Standard Equation of a Parabola: (Vertex at the origin) Equation Focus Directrix 2 x = 4py (0, p) y = –p (If the x-term is squared, the parabola is up or down) Equation 2 y = 4px Focus (p, 0) Directrix x = –p (If the y-term is squared, the parabola is left or right) Where p is the direct distance from the vertex to the focus. * note: if p < 0, then parabola opens down or left Standard Equation of a Parabola: (Vertex at (h, k) ) Equation 2 (x – h) = 4p(y - k) Focus (h, k + p) Directrix y=k–p (If the x term is squared, the parabola is up or down) Equation 2 (y - k) = 4p(x – h) Focus Directrix (h + p, k) x=h–p (If the y term is squared, the parabola is left or right) Ex 1: Determine the focus and directrix of the parabola 2 y = 4x : Since x is squared, the parabola goes up or down… 2 2 Solve for x x = 4py 2 y = 4x 4 4 2 x = 1/4y Solve for p 4p = 1/4 p = 1/16 Focus: (0, p) Directrix: y = –p Focus: (0, 1/16) Directrix: y = –1/16 Let’s see what this parabola looks like... Ex 2: Determine the focus and directrix of the parabola 2 –3y – 12x = 0 : Since y is squared, the parabola goes left or right… 2 2 Solve for y y = 4px 2 –3y = 12x 2 –3y = 12x –3 –3 2 y = –4x Solve for p 4p = –4 p = –1 Focus: (p, 0) Directrix: x = –p Focus: (–1, 0) Directrix: x = 1 Let’s see what this parabola looks like... Ex 3: Write the standard form of the equation of the parabola with focus at (0, 3) and vertex at the origin. Since the focus is on the y axis,(and vertex at the origin) the parabola goes up or down… x2 = 4py Since p = 3, the standard form of the equation is x2 = 12y Ex 4: Write the standard form of the equation of the parabola with directrix x = –1 and vertex at the origin. Since the directrix is parallel to the y axis,(and vertex at the origin) the parabola goes left or right… y2 = 4px Since p = 1, the standard form of the equation is y2 = 4x Summary… • • • • • • • • Opens Up / Down 2 (x – h) = 4p(y - k) p pos up p neg down Vertex: V (h , k) Axis: x = h Focus: F(h , k + p) Directrix: y = k – p 9_1 Parabola Examples • Opens Right / Left • (y - k)2 = 4p(x – h) • • • • • • p pos right p neg left Vertex: V (h, k) Axis: y = k Focus: F (h + p, k) Directrix: x = h – p Try It Out • Given the equations below, – What is the focus? – What is the directrix? ( x 3) ( y 2) 2 x y 4y 9 0 2 4 x 12 x 12 y 7 0 2