Graphs, Transformations, Straight Line Graphs & Circles (P1.4, 1.5 & 1.6) Flashcards
describe sketching cubic graphs
when a is +ve & when a is -ve
if p is a root of the function f(x), the graph of y=f(x) touches or crosses the x-axis at the point (p,0)
describe sketching quartic graphs
when a is +ve & when a is -ve
repeated roots touch x-axis vs distinct roots intersect x-axis
describe sketching reciprocal graphs in the form y=a/x & y=a/x^2
when a is +ve & when a is -ve
asymptotes at x=0 & y=0
see pg 66 P1 textbook
what happens when a changes on reciprocal graph
affects y-coordinate
as a increases in magnitude (either more +ve or more -ve), the graph moves ‘away’ from the x- & y-axis
describe using intersection points of graphs to solve equations
x-coordinates at the point of intersection are the solution(s) to the equation f(x)=g(x)
describe translating graphs
y = f(x) + a is translation of y=f(x) by the vector (0 a)
y = f(x+a) is translation of y=f(x) by the vector (-a 0)
when you translate a graph, any asymptotes are also translated by the same vector
describe stretching graphs
y = af(x) is a stretch of y=f(x) by scale factor a in vertical direction e.g. sf 2 means double y-coordinates
y = f(ax) is stretch of y=f(x) by scale factor 1/a in horizontal direction
y = -f(x) is a reflection of y=f(x) in the x-axis
y = f(-x) is a reflection of y=f(x) in the y-axis
transform graphs of unfamiliar functions
do Qs
what is the expression for the gradient of a line joining 2 points?
points (x1, y1) & (x2, y2)
y2 - y1 / x2 - x1
link the equation of a line, its gradient & intercept
y = mx + c
m = gradient
c = y-intercept
what is the equation of a line using gradient & one or 2 points?
for a line passing through the point (x1,y1):
y - y1 = m(x-x1)
intersection for 2 straight lines
make equations = to each other
describe gradients for parallel & perpendicular lines
parallel lines have the same gradient
for line with gradient m, the perpendicular line has the gradient -1/m
the product of their gradients is -1
what is the equation for finding the distance, d, b/w (x1, y1) & (x2, y2)?
d = √(x2-x1)^2 + (y2-y1)^2
describe modelling with straight lines
2 quantities are directly proportional when they increase at the same rate - the graph is a straight line passing through the origin
y α x is the same as y = kx
what is the expression for the midpoint of a line segment with endpoints (x1,y1) & (x2,y2)
((x1+x2)/2 , (y1+y2)/2)
what is the equation of a circle with centre (a,b) & radius r
(x-a)^2 + (y-b)^2 = r^2
describe how to find the centre & radius of a circle with an equation given in expanded form x^2 + y^2 + 2fx + 2gy + c = 0
complete the square for the x & y terms
centre (-f, -g) radius √(f^2 + g^2 -c)
a straight line can intersect a circle…
once by touching the circle, twice or not at all
tangent to circle
is perpendicular to the radius of the circle at the point of intersection
chord to circle
the perpendicular bisector of a chord will go through the centre of the circle
circumcircle of a triangle
each vertex of the triangle touches the circumference of the circle
the centre of the circle is the circumcentre & is the point where the perpendicular bisectors of each side intersect
if PRQ = 90degrees
R lies on the circle with diameter PQ
angle in a semicircle is always a right angle
how do you find the centre of the circle given any three points on the circumference?
find the equations of the perpendicular bisectors of 2 different chords
find the coordinates of the point of intersection of these perpendicular bisectors
