Stewart Precalc 6e Section 11.1: Parabolas

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Stewart Precalc 6e Section 11.1: Parabolas 0/62
1 Concepts · Level 1
A parabola is the set of all points in the plane that are equidistant from a fixed point called the (i)____ and a fixed line called the (ii)____ of the parabola.
2 Concepts · Level 1
The graph of \(x^2 = 4 p y\) is a parabola with focus \(F\) and a horizontal directrix. Express the coordinates of \(F\) and the equation of the directrix in terms of \(p\). Then identify the focus and directrix for the specific parabola \(x^2 = 12 y\).
3 Concepts · Level 1
The graph of \(y^2 = 4 p x\) is a parabola with focus \(F\) and a vertical directrix. Express the coordinates of \(F\) and the equation of the directrix in terms of \(p\). Then identify the focus and directrix for the specific parabola \(y^2 = 12 x\).
4 Concepts · Level 1
Label the focus, directrix, and vertex on the graphs given for the parabolas in Exercises 2 and 3. (a) \(x^2 = 12 y\) (b) \(y^2 = 12 x\)
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5 Skills - Matching · Level 1
Match the equation \(y^2 = 2 x\) with one of the graphs labeled I-VI. Give reasons for your answer.
6 Skills - Matching · Level 1
Match the equation \(y^2 = -\dfrac{1}{4} x\) with one of the graphs labeled I-VI. Give reasons for your answer.
7 Skills - Matching · Level 1
Match the equation \(x^2 = -6 y\) with one of the graphs labeled I-VI. Give reasons for your answer.
8 Skills - Matching · Level 1
Match the equation \(2 x^2 = y\) with one of the graphs labeled I-VI. Give reasons for your answer.
9 Skills - Matching · Level 1
Match the equation \(y^2 - 8 x = 0\) with one of the graphs labeled I-VI. Give reasons for your answer.
10 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(x^2 = 9 y\), and sketch its graph.
11 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(x^2 = y\), and sketch its graph.
12 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(y^2 = 4 x\), and sketch its graph.
13 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(y^2 = 3 x\), and sketch its graph.
14 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(y = 5 x^2\), and sketch its graph.
15 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(y = -2 x^2\), and sketch its graph.
16 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(x = -8 y^2\), and sketch its graph.
17 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(x = \dfrac{1}{2} y^2\), and sketch its graph.
18 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(x^2 + 6 y = 0\), and sketch its graph.
19 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(x - 7 y^2 = 0\), and sketch its graph.
20 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(5 x + 3 y^2 = 0\), and sketch its graph.
21 Skills - Focus/Directrix/Focal Diameter · Level 2
Find the focus, directrix, and focal diameter of the parabola \(8 x^2 + 12 y = 0\), and sketch its graph.
22 Skills - Graphing Device · Level 1
Use a graphing device to graph the parabola \(x^2 = 16 y\).
23 Skills - Graphing Device · Level 1
Use a graphing device to graph the parabola \(x^2 = -8 y\).
24 Skills - Graphing Device · Level 1
Use a graphing device to graph the parabola \(y^2 = -\dfrac{1}{3} x\).
25 Skills - Graphing Device · Level 1
Use a graphing device to graph the parabola \(8 y^2 = x\).
26 Skills - Graphing Device · Level 1
Use a graphing device to graph the parabola \(4 x + y^2 = 0\).
27 Skills - Graphing Device · Level 1
Use a graphing device to graph the parabola \(x - 2 y^2 = 0\).
28 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and focus \(F(0, 2)\).
29 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and focus \(F\left(0, -\dfrac{1}{2}\right)\).
30 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and focus \(F(-8, 0)\).
31 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and focus \(F(5, 0)\).
32 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and directrix \(x = 2\).
33 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and directrix \(y = 6\).
34 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and directrix \(y = -10\).
35 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin and directrix \(x = -\dfrac{1}{8}\).
36 Skills - Find Equation from Condition · Level 3
Find an equation for the parabola that has its vertex at the origin, with focus on the positive \(x\)-axis, and the focus \(2\) units away from the directrix.
37 Skills - Find Equation from Condition · Level 3
Find an equation for the parabola that has its vertex at the origin and whose directrix has \(y\)-intercept \(6\).
38 Skills - Find Equation from Condition · Level 2
Find an equation for the parabola that has its vertex at the origin, opens upward, with focus \(5\) units from the vertex.
39 Skills - Find Equation from Condition · Level 3
Find an equation for the parabola that has its vertex at the origin, focal diameter \(8\), and focus on the negative \(y\)-axis.
40 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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41 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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42 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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43 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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44 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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45 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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46 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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47 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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48 Skills - Find Equation from Graph · Level 2
Find an equation of the parabola whose graph is shown.
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49 Skills - Family of Parabolas · Level 3
(a) Find equations for the family of parabolas with vertex at the origin and with directrixes \(y = \dfrac{1}{2}\), \(y = 1\), \(y = 4\), and continuing in the same pattern. (b) Draw the graphs. What do you conclude?
50 Skills - Family of Parabolas · Level 3
(a) Find equations for the family of parabolas with vertex at the origin, focus on the positive \(y\)-axis, and focal diameters \(1\), \(2\), \(4\), and \(8\). (b) Draw the graphs. What do you conclude?
51 Applications - Parabolic Reflector · Level 3
A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is \(12\) cm. (a) Find an equation of the parabola. (b) Find the diameter \(d(C, D)\) of the opening, \(20\) cm from the vertex.
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52 Applications - Satellite Dish · Level 3
A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus \(F\). The reflector is \(1\) ft deep and \(20\) ft wide from rim to rim. How far is the receiver from the vertex of the parabolic reflector?
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53 Applications - Suspension Bridge · Level 3
In a suspension bridge the shape of the suspension cables is parabolic. The bridge has towers that are \(600\) m apart, and the lowest point of the suspension cables is \(150\) m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex.
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54 Applications - Reflecting Telescope · Level 4
The Hale telescope at the Mount Palomar Observatory has a \(200\)-in. mirror. The mirror is constructed in a parabolic shape that focuses light at the prime focus. The mirror is \(3.79\) in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.
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55 Discovery - Discussion - Writing · Level 2
Parabolas in the Real World. Several examples of the uses of parabolas are given in the text. Find other situations in real life in which parabolas occur. Consult a scientific encyclopedia in the reference section of your library, or search the Internet.
56 Discovery - Discussion - Writing · Level 3
Light Cone from a Flashlight. A flashlight is held to form a lighted area on the ground. Is it possible to angle the flashlight in such a way that the boundary of the lighted area is a parabola? Explain your answer.
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57 Example - Finding the Equation of a Parabola · Level 2
Find an equation for the parabola with vertex \(V(0, 0)\) and focus \(F(0, 2)\), and sketch its graph.
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58 Example - Finding the Focus and Directrix of a Parabola from Its Equation · Level 2
Find the focus and directrix of the parabola \(y = -x^2\), and sketch the graph.
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59 Example - A Parabola with Horizontal Axis · Level 2
A parabola has the equation \(6x + y^2 = 0\). *(a)* Find the focus and directrix of the parabola, and sketch the graph. *(b)* Use a graphing calculator to draw the graph.
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60 Example - Focal Diameter of a Parabola · Level 2
Find the focus, directrix, and focal diameter of the parabola \(y = \dfrac{1}{2} x^2\), and sketch its graph.
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61 Example - A Family of Parabolas · Level 3
(a) Find equations for the parabolas with vertex at the origin and foci \(F_1\left(0, \dfrac{1}{8}\right)\), \(F_2\left(0, \dfrac{1}{2}\right)\), \(F_3(0, 1)\), and \(F_4(0, 4)\). (b) Draw the graphs of the parabolas in part (a). What do you conclude?
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62 Example - Application: Searchlight Reflector · Level 3
A searchlight has a parabolic reflector that forms a bowl which is \(12\) in. wide from rim to rim and \(8\) in. deep. If the filament of the light bulb is located at the focus, how far from the vertex of the reflector is it?
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