📈College Algebra Unit 12 – Analytic Geometry

Key Concepts and Definitions

• Analytic geometry combines algebra and geometry to solve problems involving geometric shapes on a coordinate plane
• Coordinate plane consists of two perpendicular number lines (x-axis and y-axis) that intersect at the origin (0, 0)
• Points on the coordinate plane are represented by ordered pairs (x, y), where x is the horizontal distance from the origin and y is the vertical distance
• Distance formula calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint formula finds the coordinates of the midpoint $(x_m, y_m)$ between two points $(x_1, y_1)$ and $(x_2, y_2)$: $x_m = \frac{x_1 + x_2}{2}, y_m = \frac{y_1 + y_2}{2}$
• Slope of a line measures its steepness and direction, calculated as the change in y-coordinates divided by the change in x-coordinates: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Positive slope indicates a line rising from left to right, while a negative slope indicates a line falling from left to right
  • Undefined slope occurs when a line is vertical (change in x-coordinates is zero)
• Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other

Coordinate Systems and Graphing

• Cartesian coordinate system is the most common coordinate system used in analytic geometry
  • Consists of two perpendicular number lines (x-axis and y-axis) that intersect at the origin (0, 0)
  • Points are represented by ordered pairs (x, y)
• Polar coordinate system uses an angle and a distance from a fixed point (pole) to represent points
  • Points are represented by $(r, \theta)$, where $r$ is the distance from the pole and $\theta$ is the angle from the positive x-axis
• Graphing equations involves plotting points that satisfy the equation on the coordinate plane
  • Linear equations graph as straight lines
  • Quadratic equations graph as parabolas
  • Circles, ellipses, and hyperbolas have specific equations that produce their respective shapes when graphed
• Intercepts are points where a graph crosses the x-axis (x-intercept) or y-axis (y-intercept)
  • To find intercepts, set the other variable equal to zero and solve for the remaining variable
• Symmetry can be used to identify the shape and properties of a graph
  • Reflectional symmetry occurs when a graph is unchanged by reflection across a line (line of symmetry)
  • Rotational symmetry occurs when a graph is unchanged by rotation about a point (center of symmetry)

Lines and Linear Equations

• Linear equations are equations that graph as straight lines on the coordinate plane
• Slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept
  • Slope represents the change in y-coordinates divided by the change in x-coordinates
  • Y-intercept is the point where the line crosses the y-axis (when x = 0)
• Point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope
  • Useful when given a point and the slope of a line
• Standard form of a linear equation is $Ax + By = C$, where $A$, $B$, and $C$ are constants and $A$ and $B$ are not both zero
  • Can be converted to slope-intercept form by solving for y
• Parallel lines have the same slope and different y-intercepts
  • Equations of parallel lines differ only in their y-intercepts (b-values)
• Perpendicular lines have slopes that are negative reciprocals of each other
  • Product of the slopes of perpendicular lines is -1

Circles and Their Equations

• Circles are the set of all points in a plane that are equidistant from a fixed point (center)
• Standard equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$
  • Expands to $x^2 + y^2 - 2hx - 2ky + h^2 + k^2 = r^2$
• General form of a circle equation is $x^2 + y^2 + Dx + Ey + F = 0$, where $D$, $E$, and $F$ are constants
  • Can be converted to standard form by completing the square for x and y terms
• To find the center and radius of a circle from its equation:
  • Rewrite the equation in standard form by completing the square
  • Center coordinates are the negatives of the coefficients of the x and y terms, divided by 2
  • Radius is the square root of the constant term on the right side of the equation
• Circles are symmetrical about both the x-axis and y-axis, as well as about their center
• Tangent lines to a circle intersect the circle at exactly one point and are perpendicular to the radius drawn to the point of tangency

Conic Sections: Parabolas, Ellipses, and Hyperbolas

• Conic sections are curves formed by the intersection of a plane with a double cone
  • Parabolas are formed when the plane is parallel to one side of the cone
  • Ellipses are formed when the plane intersects both sides of the cone at an angle
  • Hyperbolas are formed when the plane is perpendicular to the cone's axis
• Parabolas have a single focal point and a directrix line
  • Standard equation of a parabola with vertex at the origin is $y = ax^2$ (vertical) or $x = ay^2$ (horizontal), where $a$ determines the shape and orientation
  • General equation of a parabola with vertex $(h, k)$ is $(y - k) = a(x - h)^2$ (vertical) or $(x - h) = a(y - k)^2$ (horizontal)
• Ellipses have two focal points and are defined by the sum of the distances from any point on the ellipse to the two foci
  • Standard equation of an ellipse with center at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes
  • General equation of an ellipse with center $(h, k)$ is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
• Hyperbolas have two focal points and are defined by the difference of the distances from any point on the hyperbola to the two foci
  • Standard equation of a hyperbola with center at the origin is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (horizontal) or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (vertical)
  • General equation of a hyperbola with center $(h, k)$ is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ (horizontal) or $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ (vertical)

Transformations and Rotations

• Transformations are operations that change the position, size, or shape of a graph
• Translations shift a graph horizontally or vertically without changing its shape
  • To translate a graph $h$ units horizontally and $k$ units vertically, replace $x$ with $(x - h)$ and $y$ with $(y - k)$ in the equation
• Reflections flip a graph across a line (usually the x-axis, y-axis, or line $y = x$)
  • To reflect a graph across the x-axis, replace $y$ with $-y$ in the equation
  • To reflect a graph across the y-axis, replace $x$ with $-x$ in the equation
  • To reflect a graph across the line $y = x$, swap $x$ and $y$ in the equation
• Dilations stretch or shrink a graph proportionally from a fixed point (usually the origin)
  • To dilate a graph horizontally by a factor of $a$ and vertically by a factor of $b$, replace $x$ with $\frac{x}{a}$ and $y$ with $\frac{y}{b}$ in the equation
• Rotations turn a graph about a fixed point (usually the origin) by a specified angle
  • To rotate a graph counterclockwise by angle $\theta$, replace $x$ with $x\cos\theta - y\sin\theta$ and $y$ with $x\sin\theta + y\cos\theta$ in the equation
• Compositions of transformations can be performed by applying each transformation in sequence
  • Order of transformations matters, as different orders may produce different results

Applications in Real-World Problems

• Analytic geometry has numerous applications in various fields, such as physics, engineering, and computer graphics
• Projectile motion can be modeled using parabolic equations
  • Horizontal and vertical components of motion are analyzed separately
  • Equation for the path of a projectile: $y = -\frac{g}{2v_0^2\cos^2\theta}x^2 + \tan\theta x + h$, where $g$ is acceleration due to gravity, $v_0$ is initial velocity, $\theta$ is launch angle, and $h$ is initial height
• Planetary orbits can be approximated as ellipses with the sun at one focus (Kepler's first law)
  • Eccentricity of an elliptical orbit determines how much it deviates from a circular orbit
• Reflective properties of parabolas and ellipses are used in designing satellite dishes, car headlights, and whispering galleries
  • Parabolic mirrors focus incoming light rays parallel to the axis to a single focal point
  • Elliptical mirrors have two focal points, allowing sound waves or light rays to reflect from one focus to the other
• Hyperbolic geometry is used in special relativity to describe the spacetime geometry of the universe
  • Minkowski diagrams use hyperbolas to represent the world lines of objects moving at constant velocities
• Computer graphics and animation rely on transformations and rotations to manipulate and render 2D and 3D objects
  • Affine transformations (translations, rotations, scaling, and shearing) are used to create realistic motion and deformations

Common Mistakes and How to Avoid Them

• Confusing the order of coordinates in an ordered pair (x, y)
  • Remember that x always comes first, representing the horizontal position, followed by y, representing the vertical position
• Incorrectly calculating the slope of a line
  • Make sure to use the correct order of points when using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Be careful with negative signs and fractions when simplifying the slope
• Misinterpreting the signs of coefficients in equations of circles, parabolas, ellipses, and hyperbolas
  • Pay attention to the signs of the coefficients and how they affect the shape and orientation of the graph
• Forgetting to distribute negative signs when expanding or factoring equations
  • Be thorough in applying the distributive property and keep track of negative signs
• Incorrectly applying transformations or performing them in the wrong order
  • Remember the effects of each transformation on the graph and apply them in the correct sequence
  • Be particularly careful with rotations, as they involve trigonometric functions and can be more complex than other transformations
• Neglecting to consider the domain and range of functions when graphing
  • Identify any restrictions on the input (domain) and output (range) values based on the context of the problem or the nature of the function
• Not checking the reasonableness of answers or verifying solutions graphically
  • Always double-check your work and compare your algebraic solutions to the graphical representation to ensure consistency and accuracy
• Seeking help and clarification when needed
  • Don't hesitate to ask your instructor, classmates, or tutors for assistance if you are struggling with a concept or problem
  • Utilizing available resources, such as textbooks, online tutorials, and practice problems, can help reinforce your understanding and skills in analytic geometry