Parabolas and circles are key players in the conic sections family. They're everywhere, from satellite dishes to car wheels. Understanding their shapes, equations, and real-world uses is crucial for mastering this topic.

These curves have unique features that make them special. Parabolas have a vertex and symmetry, while circles are all about the center and radius. Graphing and finding equations for both shapes involves similar steps, making them a perfect pair to study together.

Parabolas and circles: Components and characteristics

Parabola fundamentals

• A parabola is a U-shaped curve that is symmetrical and has a single turning point called the vertex
• Parabolas extend infinitely in one direction
• The vertex form of a parabola is $y=a(x-h)^2+k$, where $(h,k)$ is the vertex and $a$ determines the direction and width of the parabola
• Examples of parabolic shapes include satellite dishes and the Gateway Arch in St. Louis

Parabola symmetry and key features

• Parabolas have an axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror images
• The equation of the axis of symmetry is $x=h$
• The directrix is a horizontal line perpendicular to the axis of symmetry
• The focus is a point on the axis of symmetry that is equidistant from the vertex as the directrix

Circle fundamentals

• A circle is a round plane figure whose boundary consists of points equidistant from the center
• The standard form of a circle is $(x-h)^2+(y-k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius
• Examples of circular objects include wheels, coins, and pizzas

Circle symmetry and key features

• Circles are symmetrical about any diameter, a line segment that passes through the center and has its endpoints on the circle
• The chord is a line segment that connects any two points on the circle's circumference
• The central angle is an angle formed by two radii drawn to the endpoints of an arc
• The inscribed angle is an angle formed by two chords that share an endpoint on the circle's circumference

Graphing parabolas and circles

Graphing parabolas

• To graph a parabola, plot the vertex $(h,k)$, then plot additional points on either side of the vertex using the equation
• Connect the plotted points to form the U-shape
• The sign of $a$ in the vertex form determines if the parabola opens up $(a>0)$ or down $(a<0)$
• The absolute value of $a$ affects the width of the parabola (larger $|a|$ means narrower parabola)

Parabola graphing techniques

• Graphing parabolas requires a t-chart to solve for ordered pairs, with x-values symmetrical around the axis of symmetry
• Choose x-values that are equidistant from $h$, such as $h-1$, $h$, and $h+1$, to ensure symmetry
• Substitute the x-values into the equation to find the corresponding y-values
• Plot the ordered pairs and connect them to form the parabola

Graphing circles

• To graph a circle, plot the center $(h,k)$, then plot four points $r$ units up, down, left, and right from the center
• Connect the four plotted points to form the circle
• The general form of a circle is $x^2+y^2+Ax+By+C=0$
• To graph a circle in general form, convert it to standard form by completing the square for both $x$ and $y$

Circle graphing techniques

• Identify the coefficients $A$ and $B$ of the $x$ and $y$ terms, respectively
• Divide $A$ and $B$ by 2 and square the results to find $h$ and $k$
• Substitute $h$ and $k$ into the general form and simplify to find $C$
• Use the Pythagorean theorem to solve for the radius: $r=\sqrt{h^2+k^2-C}$
• Plot the center and points, then connect to form the circle

Equations of parabolas and circles

Determining parabola equations

• To find the equation of a parabola, identify the vertex $(h,k)$ and a point $(x,y)$ on the curve
• Substitute the coordinates into $y=a(x-h)^2+k$ and solve for $a$
• The axis of symmetry and directrix can also be used to determine the equation of a parabola
• Given the focus $(h,k+p)$ and directrix $y=k-p$, the equation is $y=\frac{1}{4p}(x-h)^2+k$

Determining circle equations

• To find the equation of a circle, identify the center $(h,k)$ and radius $r$
• Substitute the values into $(x-h)^2+(y-k)^2=r^2$
• Given the general form $x^2+y^2+Ax+By+C=0$, convert to standard form to identify the center and radius
• The equation of a circle can also be determined using the midpoint and distance formulas if given the endpoints of a diameter

Converting between forms

• To convert from standard form to general form, expand the squared binomials and simplify
• To convert from general form to standard form, complete the square for both $x$ and $y$
• Vertex form is specific to parabolas and is not used for circles
• Converting between forms is often necessary to extract relevant information for problem-solving

Real-world applications of parabolas and circles

Parabola applications

• Many real-world situations can be modeled by parabolas, such as trajectories of objects (footballs, basketballs), satellite dishes, suspension bridges, and certain curves in nature
• The vertex often represents the maximum or minimum value of the parabola in a particular context
• Example: The height $h$ of a ball thrown upward with an initial velocity of 30 ft/s from a height of 5 ft can be modeled by $h=-16t^2+30t+5$, where $t$ is time in seconds
• The vertex $(0.9375, 19.0625)$ represents the maximum height of the ball at 0.9375 seconds after being thrown

Circle applications

• Circles can be used to model the wheels of a car, traffic roundabouts, center-pivot irrigation systems, and the distance from a central location
• Example: A center-pivot irrigation system rotates around a central point, watering a circular area. If the system is 1,320 feet long, find the area it irrigates.
• The area can be found using $A=\pi r^2$, where $r=1,320$. The system irrigates approximately 5,471,136 square feet.

Problem-solving strategies

• Real-world problems often require setting up an equation based on given information and constraints, then solving that equation for a specific value
• Identify the key components of the problem, such as the vertex, center, radius, or points on the curve
• Determine which form of the equation (vertex, standard, general) is most appropriate for the given information
• Translate the problem into an equation by substituting the known values
• Solve the equation for the desired quantity, such as a coordinate, dimension, or optimum value
• Parabolas and circles may need to be translated between different forms to extract the relevant information to answer the question