HW: Pg ,28-30even, even, 50-56even, Do Now: Take out your pencil, notebook, and calculator. 1) Draw a parabola with a vertex, focus and directrix. Objectives: You will be able to define parametric equations, graph curves parametrically, and solve application problems using parametric equations. Agenda: 1.Do Now 2. Graphing parabolas with a directrix and focus not at the vertex. Precalculus April 9, 2015

8.1 Graph and Write Equations of Parabolas day 2 What does it mean if a parabola has a translated vertex? What general equations can you use for a parabola when the vertex has been translated?

Standard Equation of a Translated Parabola Vertical axis: vertex: (h, k) focus: (h, k + p) directrix: y = k – p axis of symmetry: x = h ( x − h) 2 = 4p(y − k)

Standard Equation of a Translated Parabola Horizontal axis: vertex: (h, k) focus: (h + p, k) directrix: x = h - p axis of symmetry: y = k (y − k) 2 = 4p(x − h)

Example 1 Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. Sketch the info. The parabola opens downward, so the equation is of the form vertex: (-3,3) h = -3, k = 3 p = -1 (x − h) 2 = 4p(y − k) (x + 3) 2 = 4(−1)(y − 3) Focus (h, k+p) (-3, 3+p)=(-3, 2)

Graph (x – 2) 2 = 8 (y + 3). SOLUTION STEP 1 Compare the given equation to the standard form of an equation of a parabola. You can see that the graph is a parabola with vertex at (2, – 3), focus (2, – 1) and directrix y = – 5 Draw the parabola by making a table of value and plot y point. Because p > 0, the parabola open to the right. So use only points x- value x12345 y–2.875–3– 2.875–2.5– STEP 3 Draw a curve through the points. STEP 2

Example 4 Write an equation of a parabola whose vertex is at ( −2,1) and whose focus is at (−3, 1). Begin by sketching the parabola. Because the parabola opens to the left, it has the form (y −k)2 = 4p(x − h) Find h and k: The vertex is at ( − 2,1) so h = − 2 and k = 1 Find p: The distance between the vertex (−2,1) and the focus (−3,1) by using the distance formula. p = −1 (y − 1) 2 = −4(x + 2)

Write an equation of the parabola whose vertex is at (– 2, 3) and whose focus is at (– 4, 3). SOLUTION STEP 1 Determine the form of the equation. Begin by making a rough sketch of the parabola. Because the focus is to the left of the vertex, the parabola opens to the left, and its equation has the form (y – k) 2 = 4p(x – h) where p < 0. STEP 2 Identify h and k. The vertex is at (– 2, 3), so h = – 2 and k = 3. STEP 3 Find p. The vertex (– 2, 3) and focus (  4, 3) both lie on the line y = 3, so the distance between them is p | = | – 4 – (– 2) | = 2, and thus p = +2. Because p < 0, it follows that p = – 2, so 4p = – 8. The standard form of the equation is (y – 3) 2 = – 8(x + 2).

Write the standard form of a parabola with vertex at (3, – 1) and focus at (3, 2). SOLUTION STEP 1Determine the form of the equation. Begin by making a rough sketch of the parabola. Because the focus is to the left of the vertex, the parabola opens to the left, and its equation has the form (x – h) 2 = 4p(y – k) where p > 0. STEP 2 Identify h and k. The vertex is at (3,– 1), so h = 3 and k = –1. STEP 3 Find p. The vertex (3, – 1) and focus (3, 2) both lie on the line x = 3, so the distance between them is p | = | – 2 – (– 1) | = 3, and thus p = + 3. Because p > 0, it follows that p = 3, so 4p = 12. The standard form of the equation is ( x – 3) 2 = 12(y + 1)

Graphing a parabola on your calculator

Using standard form

On the sidelines of each televised football game, ESPN uses a parabolic reflector with a microphone at the reflector’s focus to capture the conversations among players on the field. If the parabolic reflector is 3ft across and 1 ft deep, where should the microphone be placed?

What does it mean if a parabola has a translated vertex? It means that the vertex of the parabola has been moved from (0,0) to (h,k). What general equations can you use for a parabola when the vertex has been translated? (y-k) 2 =4p(x-h) (x-h) 2 =4p(y-k)