§ 10.4 the parabola; identifying conic sections. blitzer, intermediate algebra, 5e – slide #2...
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§ 10.4
The Parabola; Identifying Conic Sections
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 10.4
Equation of a Parabola
We looked at parabolas in Chapter 8, viewing them as graphs of quadratic functions. In this section, we return again to the study of parabolas. Within this chapter on Conics, we extendour study by considering the formal definition of the parabola.
A parabola is the set of all points in the plane that are equidistant from a fixed line (called the directrix) and a fixed point (called the focus).
Note that the definition of both the hyperbola and the ellipse involved two fixed points, the foci. By contrast, the definition of a parabola is based on one point and a line.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 10.4
Equation of a Parabola
Definition of a ParabolaA parabola is the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus) that is not on the line.
Directrix
FocusParabola Vertex
Axis of Symmetry
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 10.4
Equation of a Parabola
Parabolas Opening to the Left or to the RightThe graphs of are parabolas opening to the left or to the right.
1) If a > 0, the graph opens to the right. If a < 0, the graph opens to the left.
2) The vertex of
3) The y-coordinate of the vertex of
cbyayxhkyax 22 and
. is 2 h, khkyax
.2
is 2
a
bycbyayx
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 10.4
Equation of a Parabola
CONTINUECONTINUEDD
(h, k) y = k
0
2
a
hkyax
x
y
(h, k)y = k
0
2
a
hkyax
x
y
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 10.4
Equation of a Parabola
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Graph:
We can graph this equation by following the steps in the preceding box. We begin by identifying values for a, k, and h.
.12 2 yx
1) Determine how the parabola opens. Note that a, the coefficient of , is 1. Thus, a > 0; this positive value tells us that the parabola opens to the right.
12 2 yx
hkyax 2
2y
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 10.4
Equation of a Parabola
Graphing Horizontal ParabolasTo graph
1) Determine whether the parabola opens to the left or to the right. If a > 0, it opens to the right. If a < 0, it opens to the left.
2) Determine the vertex of the parabola. The vertex of is
(h, k). The y-coordinate of the vertex of
Substitute this value of y- into the equation to find the x-coordinate.
3) Find the x-intercept by replacing y with 0.
4) Find any y-intercepts by replacing x with 0. Solve the resulting quadratic equation for y.
5) Plot the intercepts and the vertex. Connect a more accurate graph, select values for y on each side of the axis of symmetry and compute values for x.
,or 22 cbyayxhkyax
hkyax 2
.2
is 2
a
bycbyayx
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 10.4
Equation of a Parabola
2) Find the vertex. The vertex of the parabola is at (h, k). Because k = 2 and h = 1, the parabola has its vertex at (1, 2).
3) Find the x-intercept. Replace y with 0 in
51412120 22 x
CONTINUECONTINUEDD
.12 2 yx
The x-intercept is 5. The parabola passes through (5, 0).
4) Find the y-intercepts. Replace x with 0 in the given equation. 12 2 yx This is the given equation.
120 2 y Replace x with 0.
221 y Subtract 1 from both sides.This equation clearly has no solutions since the left side is a negative number. Therefore, there are no y-intercepts.
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 10.4
Equation of a Parabola
CONTINUECONTINUEDD
5) Graph the parabola. With a vertex at (1, 2), an x-intercept at 5, and no y-intercepts, the graph of the parabola is shown as follows. The axis of symmetry is the horizontal line whose equation is y = 2.
-2
-1
0
1
2
3
4
5
6
0 2 4 6 8 10 12
Vertex: (1, 2)
x-intercept: (5, 0)
Axis of symmetry: y = 2.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 10.4
Equation of a Parabola
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Graph:
1) Determine how the parabola opens. Note that a, the coefficient of , is -2. Thus a < 0; this negative value tells us that the parabola opens to the left.
.342 2 yyx
2) Find the vertex. We know that the y-coordinate of the vertex
is . We identify a, b, and c ina
by
2
2y
a = -2
342 2 yyx
b = 4 c = -3
.2 cbyayx
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 10.4
Equation of a Parabola
Substitute the values of a and b into the equation for the y-coordinate:
The y-coordinate of the vertex is 1. We substitute 1 for y into the parabola’s equation, to find the x-coordinate:
.122
4
2
a
by
342 2 yyx
CONTINUECONTINUEDD
.134231412 2 x
The vertex is at (-1, 1).
3) Find the x-intercept. Replace y with 0 in .342 2 yyx
330402 2 x
The x-intercept is -3. The parabola passes through (-3, 0).
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 10.4
Equation of a Parabola
CONTINUECONTINUEDD
4) Find the y-intercepts. Replace x with 0 in the given equation.
342 2 yyx This is the given equation.3420 2 yy Replace x with 0.
22
32444 2
yUse the quadratic formula to solve for y.
4
24164
y Simplify.
4
84
y Subtract.
This equation clearly has no solutions since the radicand is a negative number. Therefore, there are no y-intercepts.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 10.4
Equation of a Parabola
CONTINUECONTINUEDD
5) Graph the parabola. With a vertex at (-1, 1), an x-intercept at -3, and no y-intercepts, the graph of the parabola is shown below. The axis of symmetry is the horizontal line whose equation is y = 1.
-3
-2
-1
0
1
2
3
4
5
-25 -20 -15 -10 -5 0
Vertex: (-1, 1)
x-intercept: (-3, 0)
Axis of symmetry: y = 1.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 10.4
Equations of Conic Sections
Recognizing Conic Sections from EquationsConic Section How to Identify the Equation Example
Circle When - and -terms are on the same side, they have the same coefficient.
Ellipse When - and -terms are on the same side, they have different coefficients of the same sign.
or (dividing by 64)
Hyperbola When - and -terms are on the same side, they have coefficients with opposite signs.
or (dividing by 36)
Parabola Only one of the variables is squared.
2y2x
2y2x
2y2x
1622 yx
64164 22 yx
1416
22
yx
3649 22 xy
194
22
xy
542 yyx
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 10.4
Equations of Conic Sections
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola:
(Throughout the solution, in addition to identifying each equation’s graph, we’ll also discuss the graph’s important features.) If both variables are squared, the graph of the equation is not a parabola. In both cases, we collect the - and -terms on the same side of the equation.
.3273(b)436(a) 2222 yxyx
2y2x
22 436(a) yx The graph cannot be a parabola. To see if it is a circle, an ellipse, or a hyperbola, we collect the - and -terms on the same side. Add to both sides. We obtain the following:
2y2x24y
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 10.4
Equations of Conic Sections
.364 22 yx
Because the - and -terms have different coefficients of the same sign, the equation’s graph is an ellipse.
2y2x
CONTINUECONTINUEDD
22 3273(b) yx
The graph cannot be a parabola. To see if it is a circle, an ellipse, or a hyperbola, we collect the - and -terms on the same side. Subtract from both sides. We obtain:
2y2x23y
2733 22 yx
Because the - and -terms have coefficients with opposite signs, the equation’s graph is a hyperbola.
2y2x
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 10.4
The Parabola
A parabola is the set of all points that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line.
The line passing through the focus and perpendicular to the directrix is the axis of symmetry. The point of intersection of the parabola with its axis of symmetry is the vertex.
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