Graphs, Transformations, Straight Line Graphs & Circles (P1.4, 1.5 & 1.6)

Question 1

describe sketching cubic graphs

Answer 1

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)

Question 2

describe sketching quartic graphs

Answer 2

when a is +ve & when a is -ve
repeated roots touch x-axis vs distinct roots intersect x-axis

Question 3

describe sketching reciprocal graphs in the form y=a/x & y=a/x^2

Answer 3

when a is +ve & when a is -ve
asymptotes at x=0 & y=0
see pg 66 P1 textbook

Question 4

what happens when a changes on reciprocal graph

Answer 4

affects y-coordinate
as a increases in magnitude (either more +ve or more -ve), the graph moves ‘away’ from the x- & y-axis

Question 5

describe using intersection points of graphs to solve equations

Answer 5

x-coordinates at the point of intersection are the solution(s) to the equation f(x)=g(x)

Question 6

describe translating graphs

Answer 6

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

Question 7

describe stretching graphs

Answer 7

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

Question 8

transform graphs of unfamiliar functions

Answer 8

do Qs

Question 9

what is the expression for the gradient of a line joining 2 points?

Answer 9

points (x1, y1) & (x2, y2)
y2 - y1 / x2 - x1

Question 10

link the equation of a line, its gradient & intercept

Answer 10

y = mx + c
m = gradient
c = y-intercept

Question 11

what is the equation of a line using gradient & one or 2 points?

Answer 11

for a line passing through the point (x1,y1):
y - y1 = m(x-x1)

Question 12

intersection for 2 straight lines

Answer 12

make equations = to each other

Question 13

describe gradients for parallel & perpendicular lines

Answer 13

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

Question 14

what is the equation for finding the distance, d, b/w (x1, y1) & (x2, y2)?

Answer 14

d = √(x2-x1)^2 + (y2-y1)^2

Question 15

describe modelling with straight lines

Answer 15

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

Question 16

what is the expression for the midpoint of a line segment with endpoints (x1,y1) & (x2,y2)

Answer 16

((x1+x2)/2 , (y1+y2)/2)

Question 17

what is the equation of a circle with centre (a,b) & radius r

Answer 17

(x-a)^2 + (y-b)^2 = r^2

Question 18

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

Answer 18

complete the square for the x & y terms
centre (-f, -g) radius √(f^2 + g^2 -c)

Question 19

a straight line can intersect a circle…

Answer 19

once by touching the circle, twice or not at all

Question 20

tangent to circle

Answer 20

is perpendicular to the radius of the circle at the point of intersection

Question 21

chord to circle

Answer 21

the perpendicular bisector of a chord will go through the centre of the circle

Question 22

circumcircle of a triangle

Answer 22

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

Question 23

if PRQ = 90degrees

Answer 23

R lies on the circle with diameter PQ
angle in a semicircle is always a right angle

Question 24

how do you find the centre of the circle given any three points on the circumference?

Answer 24

find the equations of the perpendicular bisectors of 2 different chords
find the coordinates of the point of intersection of these perpendicular bisectors