UGA - Math 1113E - Arnold - Fall 2024 with 100- correct answers

distance formula d = √[( x₂ - x₁)² + (y₂ - y₁)²] pythagorean theorem a²+b²=c² area of a triangle A=1/2bh midpoint formula (x₁+x₂)/2, (y₁+y₂)/2 parabola the graph of a quadratic function axis of a parabola the line passing through the focus and the vertex vertex of the parabola the highest or lowest point on the graph (h,k) = (x,y) x-intercepts X intercept: (#, 0) let y = 0, then solve for x y-intercepts Y intercept: (0. #) let x = 0, then solve for y y-axis symmetry replace x with -x (x,y) = (-x,y) reflected over y-axis x-axis symmetry replace y with -y (x,y) = (x,-y) reflected over x-axis origin symmetry replace x and y with -x and -y (x,y) = (-x,-y) reflected at an angle from quadrant I to III standard equation of a circle (x - h)^2 + (y - k)^2 = r^2 center: (h, k) radius: r point (x,y): lies on the circle completing the square ex. x^2 + y^2 - 18x + 12y -4 = 0 1. put x and y terms together on one side x^2 - 18x + y^2 + 12y = 4 2. need coefficients to = 1 (x^2 - 18x + ?) + (y^2 + 12y + ?) = 4 + ? + ? 3. take 1/2 of middle coefficient (x^2 - 18x + 9^2) + (y^2 + 12y + 6^2) = 4 + 9^2 + 6^2 (x^2 - 18x + 81) + (y^2 + 12y + 36) = 121 4. sign is the 1st sign from the equation; the number is from the 1/2 number (x - 9)^2 + (y + 6)^2 = √121 = 11 slope of a line (m) y2-y1/x2-x1 rise/run or change in y/change in x equation of a horizontal line y=b where b is the y-intercept (0,b) slope is 0 equation of a vertical line x=a where a is the x-intercept (a,0) slope is undefined point-slope form y - y1 = m(x - x1) slope-intercept form y=mx+b where m is the slope and b is the y-intercept of the line. general form for the equation of a line ax+by= c set ax=c and by=c to find the x and y intercept slope of parallel lines same slope slope of perpendicular lines opposite reciprocal (fit the fraction and make negative) function a relation in which each element of the domain (x-values) is paired with exactly one element of the range (y-values) f(x) is y vertical line test - a test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation - can not have multiple x values horizontal line test a test used to determine if the inverse of a relation is a function or one-to-one [ left point or bottom point; closed point ] right point or top point; closed point ( left point or bottom point; open point ) right point or top point; open point domain and range the x values are domain and the y values are the range domain: set all x's equal to 0 range: increasing function a graph that has a positive slope and shows growth from left to right decreasing function a graph that has a negative slope and shows a decrease from left to right constant function a function like f(x) = y = 5 that is the same regardless of the independent variable; graph is horizontal line linear function a function in which the graph of the solutions