C1 Flashcards
Function f(x)=ax^2 +bx+c
a<0?
-
Maximum turning point
Sad face ^
Function f(x)=ax^2 +bx+c
a>0?
+
Minimum turning point U
Vertex
Turning point
A line of symmetry can be drawn
Vertex= (1,3)
Line of symmetry?
X=1
Discriminant
b^2 - 4ac
How to factories a complicated
ax^2 + bx + c
- make equation so that a is +
- a X c
- find two factors of ac that when added= b (call these zx and yx)
- (ax^2 +zx)+(yx+c)
- simplify
- delete one of the like brackets make a second bracket from what you removed
- now add - negatives so you get original equation when multiplied
Completing the square
Of x^2 + bx
(X+b/2)^2 -(b/2)^2
Completing the square
Of x^2 + bx +c
(X+b/2)^2 -(b/2)^2 +c
Completing the square
3x^2 + 2x +1
Same if a is a -
3(x^2 +2/3x) +1 3((x+1/3)^2 -(1/3)^2 ) +1 3((x+1/3)^2 -(1/9))+1 3(x+1/3)^2 - 1/3 +1 3(x+1/3)^2 +2/3
Least value of the square
Vertex
Make square =0
Put in “when = x”
Whatever it makes is the least value
(X,lv)
Vertex
2(x + 1)^2 -3
(-1,-3)
-b,c
what does the C value of the completed square form show Maximum/minimum of a parabola
it is the Maximum/minimum of a parabola
When solving an equation by completing the square what must you not forget
There are two answers
One is +sqr rt
Other is -sqr rt
Quadratic formula
(-b+- /b^2 -4ac) /2a
b^2 -4ac <0
Real roots?
None
b^2 -4ac =0
1
b^2 -4ac >0
2
Equation has one real root solve to find an imbedded letter
One real root so =0
b^2 -4ac
Solve to find letter
When you square root what does it give
A + and a -
The degree of a polynomial
The highest power of x
Roots of f(x)=(2x-1)(x-2)(3x+1)
1/2 2 -1/3
Cubic functions a>0
Progresses from low left to high right
Cubic functions a<0
Progresses from high left to low right
How to draw a cubic graph
Find the 3 values of x
Find if a is a + or - (a<>0)so you know which direction it travels
Circle equation
X^2 + y^2 =r^2
(X-a)^2 + (y-b)^2 =r^2
Effect of changing a
(X-a)^2
From side to side
If it is -2 then it will touch x axis at 2
If it is 3 then it will touch x axis at -3
Effect of changing a
X^2 + c
Up and down (it is the y intercept)
translation (1/3) is applied to y=x^2
Outcome
Y=(x-1)^2 +3
Sketch the circle (x-2)^2 + (y+1)^2 =4
Translate by (2/-1)
What is the translation of a graph
The same as its vertex
Lines are perpendicular if
Both gradients multiplied =-1
m1 x m2 = -1
Lines are parallel
If gradients are the same
m1 = m2
The perpendicular line to line with gradient m
-1/m
Midpoint of a line
(X1+X2/2 ,y1+y2/2)
Distance between two points
AB =sqrt((x2-x1)^2 + (y2-y1)^2)
Gradient of a line
AB = y2-y1/x2-x1
Y=3x-2
Gradient and y intercept?
Gradient 3
Y intercept (0,-2)
Equation of a line using gradient formula
y-y1=m(x-x1)
Fining the point of intersection
Simultaneous equations
Find x and y
Put into form ax+by+c=0
Rearrange
Multiply by fraction denominators
Write as y=mx+c
Divide by coefficient of y
a perpendicular line has a
minus reciprocal gradient m becomes -1/m
parallel lines have
the same gradient
equation of a circle
(x-a)^2 + (y-b)^2 =r^2
equation of a circle center (0,0)
x^2 + y^2 =r^2
find the radius and center of circle
(x+3)^2 + (y-4)^2 -25
radius =5
center (-3, 4)
After completing the square to find the equation of a circle what is the number outside of the brackets
r^2 (and not just r)
How do you know if an angle in a circle is a right angle
If it is subtended at the circumference of a circle by a diameter.
An angle in a semicircle is a right angle
What is a tangent to a circle
a line on the outside, touches the circle edge once (has one root)
What is a normal to a circle
Line crossing through the Circle, is the normal to a tangent so splits the circle in half
at a stationary point what is dy/dx
0
the discriminant
b^2 -4ac
b^2-4ac >0
two real roots
b^2 -4ac <0
no real roots
b^2-4ac =0
one real root
the gradient of a line can be found
by differentiation
the differentiated form is also called
the derivative
the increasing function
dy/dx >0
the decreasing function
dy/dx<0
what do you do if you get 0 for d^2/dx^2 when you need to find if it is decreasing or increasing
put in -1, 0, 1 into dy/dx
you can tell what shape the graph is making
min is U max is A
another word for the gradient
rate of change
