10.3 The parabola  (Page 6/11)

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Key equations

 Parabola, vertex at origin, axis of symmetry on x -axis ${y}^{2}=4px$ Parabola, vertex at origin, axis of symmetry on y -axis ${x}^{2}=4py$ Parabola, vertex at $\text{\hspace{0.17em}}\left(h,k\right),$ axis of symmetry on x -axis ${\left(y-k\right)}^{2}=4p\left(x-h\right)$ Parabola, vertex at $\text{\hspace{0.17em}}\left(h,k\right),$ axis of symmetry on y -axis ${\left(x-h\right)}^{2}=4p\left(y-k\right)$

Key concepts

• A parabola is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and the x -axis as its axis of symmetry can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens right. If $\text{\hspace{0.17em}}p<0,$ the parabola opens left. See [link] .
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and the y -axis as its axis of symmetry can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens up. If $\text{\hspace{0.17em}}p<0,$ the parabola opens down. See [link] .
• When given the focus and directrix of a parabola, we can write its equation in standard form. See [link] .
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ and axis of symmetry parallel to the x -axis can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens right. If $\text{\hspace{0.17em}}p<0,$ the parabola opens left. See [link] .
• The standard form of a parabola with vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ and axis of symmetry parallel to the y -axis can be used to graph the parabola. If $\text{\hspace{0.17em}}p>0,$ the parabola opens up. If $\text{\hspace{0.17em}}p<0,$ the parabola opens down. See [link] .
• Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See [link] .

Verbal

Define a parabola in terms of its focus and directrix.

A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

If the equation of a parabola is written in standard form and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is positive and the directrix is a vertical line, then what can we conclude about its graph?

If the equation of a parabola is written in standard form and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is negative and the directrix is a horizontal line, then what can we conclude about its graph?

The graph will open down.

What is the effect on the graph of a parabola if its equation in standard form has increasing values of $\text{\hspace{0.17em}}p\text{?}$

As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?

The distance between the focus and directrix will increase.

Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.

${y}^{2}=4-{x}^{2}$

$y=4{x}^{2}$

yes $\text{\hspace{0.17em}}y=4\left(1\right){x}^{2}$

$3{x}^{2}-6{y}^{2}=12$

${\left(y-3\right)}^{2}=8\left(x-2\right)$

yes $\text{\hspace{0.17em}}{\left(y-3\right)}^{2}=4\left(2\right)\left(x-2\right)$

${y}^{2}+12x-6y-51=0$

For the following exercises, rewrite the given equation in standard form, and then determine the vertex $\text{\hspace{0.17em}}\left(V\right),$ focus $\text{\hspace{0.17em}}\left(F\right),$ and directrix of the parabola.

$x=8{y}^{2}$

${y}^{2}=\frac{1}{8}x,V:\left(0,0\right);F:\left(\frac{1}{32},0\right);d:x=-\frac{1}{32}$

$y=\frac{1}{4}{x}^{2}$

$y=-4{x}^{2}$

${x}^{2}=-\frac{1}{4}y,V:\left(0,0\right);F:\left(0,-\frac{1}{16}\right);d:y=\frac{1}{16}$

$x=\frac{1}{8}{y}^{2}$

$x=36{y}^{2}$

${y}^{2}=\frac{1}{36}x,V:\left(0,0\right);F:\left(\frac{1}{144},0\right);d:x=-\frac{1}{144}$

$x=\frac{1}{36}{y}^{2}$

${\left(x-1\right)}^{2}=4\left(y-1\right)$

${\left(x-1\right)}^{2}=4\left(y-1\right),V:\left(1,1\right);F:\left(1,2\right);d:y=0$

${\left(y-2\right)}^{2}=\frac{4}{5}\left(x+4\right)$

${\left(y-4\right)}^{2}=2\left(x+3\right)$

${\left(y-4\right)}^{2}=2\left(x+3\right),V:\left(-3,4\right);F:\left(-\frac{5}{2},4\right);d:x=-\frac{7}{2}$

${\left(x+1\right)}^{2}=2\left(y+4\right)$

${\left(x+4\right)}^{2}=24\left(y+1\right)$

${\left(x+4\right)}^{2}=24\left(y+1\right),V:\left(-4,-1\right);F:\left(-4,5\right);d:y=-7$

how to understand calculus?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
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can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
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This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
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