Transcript Quadratic_graphs

A graph formed by a quadratic equation will
look something like,
or
A graph with a
positive x2
A graph with a
negative x2
The simplest quadratic to graph is
y = x2
If an equation has a number added or subtracted
to it then the graph moves up or down
y = x2 + 3
y = x2 – 3
If there is a number in brackets with the x,
move the graph left or right in the opposite
direction (positive = left, negative = right)
y = (x + 3)2
y = (x – 3)2
A number higher than one in front of the x will
make the graph taller and skinnier
y = 2x2
Normally a quadratic graph follows the pattern:
Start at the lowest point. Then go
1 across, 1 up
1 across, 3 up
1 across, 5 up
1 across, 7 up …
If there is a number in front of the x2 multiply
each of the up numbers.
The graph is the same on both sides.
e.g.
1
1
1
1
y = 3x2
3 up
across, __
9 up
across, __
across, 15
__ up
21 up …
across, __
e.g.
1
1
1
1
y = -2 x2
-2 up
across, __
-6 up
across, __
across,-10
__ up
across,-14
__ up …
e.g.
1
1
1
1
y = -2 x2
2 down
across, __
6 down
across, __
across, 10
__ down
across, 14
__ down …
A negative in front of the x makes the graph
upside down.
y = -x2
1.
2.
3.
4.
5.
6.
7.
y
y
y
y
y
y
y
= x2 + 4
= x2 – 1
= (x + 2)2
= (x – 5)2
= (x + 1)2 – 2
= 3x2
=-0.5x2 + 2
Sometimes you will need to graph a qadratic
that has two brackets.
1.
2.
3.
4.
Factorise (if you need to)
Mark in the x intercepts
Find the middle point
Draw a line
Sometimes a question will be factorised,
sometimes it will not.
A quadratic that has been factorised is much
easier to graph.
y = x2 + 2x - 3
y = (x – 1)(x + 3)
The x intercepts are where the graph crosses
the x axis. To find them you make the
equation equal zero.
y = (x – 1)(x + 3)
(x – 1)(x + 3) = 0
x+3=0
x–1=0
x=0–3
x=0+1
or x = -3
x=1
Mark these points on the x axis of the graph.
x = 1 or x = -3
×
×
The next step is to find the x and y of the point
halfway between the two intercepts.
To find the x value, add the intercepts and divide
by 2.
y = (x – 1)(x + 3)
x = 1 or x = -3
1 + -3 = -2
-2 ÷ 2 = -1
The middle point of the graph is when x = -1
To find the y value of the middle point put the x
back into the formula.
y = (x – 1)(x + 3)
x = -1
y = (-1
x – 1)(-1
x + 3)
y = (-2)(2)
y = –4
Mark this point on the x axis of the graph.
x = –1
y = –4
×
×
×
Draw a curve
×
×
×
Sometimes a question will be factorised,
sometimes it will not.
A quadratic that has been factorised is much
easier to graph.
y = x2 + 8x + 12
y = (x + 2)(x + 6)
The x intercepts are where the graph crosses
the x axis. To find them you make the
equation equal zero.
y = (x + 2)(x + 6)
(x + 2)(x + 6) = 0
x+6=0
x+2=0
x=0–6
x=0-2
or x = -6
x = -2
Mark these points on the x axis of the graph.
x = -2 or x = -6
× ×
The next step is to find the x and y of the point
halfway between the two intercepts.
To find the x value, add the intercepts and divide
by 2.
y = (x + 2)(x + 6)
x = -2 or x = -6
-2 + -6 = -8
-8 ÷ 2 = -4
The middle point of the graph is when x = -4
To find the y value of the middle point put the x
back into the formula.
y = (x + 2)(x + 6)
x = -4
y = ( -4
x + 2)(-4
x + 6)
y = (-2)(2)
y = –4
Mark this point on the x axis of the graph.
x = –4
y = –4
× ×
×
Draw a curve
× ×
×
If you want an extra point to make drawing the
line a little easier multiply the numbers in the
brackets.
This is where the line crosses the y axis (the y
intercept).
Sometimes a question will be factorised,
sometimes it will not.
A quadratic that has been factorised is much
easier to graph.
y = x2 - 8x + 15
y = (x - 3)(x - 5)
The x intercepts are where the graph crosses
the x axis. To find them you make the
equation equal zero.
y = (x - 3)(x - 5)
(x - 3)(x - 5) = 0
x-5=0
x-3=0
x=0+5
x=0+3
or x = 5
x=3
Mark these points on the x axis of the graph.
x = 3 or x = 5
××
The next step is to find the x and y of the point
halfway between the two intercepts.
To find the x value, add the intercepts and divide
by 2.
y = (x - 3)(x - 5)
x = 3 or x = 5
3+ 5 = 8
8÷2=4
The middle point of the graph is when x = 4
To find the y value of the middle point put the x
back into the formula.
y = (x - 3)(x - 5)
x=4
y = (4
x - 3)(4
x - 5)
y = (1)(-1)
y = –1
Mark this point on the x axis of the graph.
x=4
y = -1
×××
Draw a curve
×××
1.
y = x2 + 4x + 3
2.
y = x2 - 2x - 8
3.
y = x2 - 10x + 21
4.
y = x2 - 9
5.
y = 3x2 - 12x + 9