Download quadratic function

Quadratic Functions – Intro
LINEAR
(review)
equations look like
slope y-intercept form
y = mx + b
e.g. y = 2x + 7
standard form
Ax + By + C = 0
e.g. 2x – y + 7 = 0
other forms: there are other ways to write linear equations…
graphs look like
y = 2x + 1
(1, 3)
(0, 1)
note: graphs are always straight lines
rise=2
run=1
table of values look like
y = 2x + 1
x
y
first differences
0 1
3–1=2
1 3
5–3=2
2 5
7–5=2
3 7
9–7=2
4 9
note: first differences are constant (don’t change)
(i.e. slopes are constant)
Quadratic Functions – Intro
note: this is the same quadratic,
just written 2 different ways
QUADRATIC:
equations look like
standard form
y = ax2 + bx + c
y = x2 – 7x + 10
factored form
y = a(x – s)(x – t)
y = (x – 2)(x – 5)
note: there are other ways to write quadratic equations…
graphs look like
note: graphs are always parabolas (“U” shaped)
8
12
4
fx  =
10
0.75x-42 -3
vertex (4, 2)
x-intercept (2, 0)
(one root is 2)
y-intercept (0, 9)
axis of symmetry
x=4
6
2
so, maximum value = 2
so, max value = 2
-5
5
10
6
15
4
x-intercept (6, 0)
-2
y-intercept (0, –6)
axis of symmetry
x=4
8
(one root is 6)
x-intercept (6, 0)
(one root is 6)
2
-4
-15
-10
-5
5
10
-6
x-intercept (2, 0)
-2
(one root is 2)
-8
vertex (4, –3)
-4
so, minimun value = –3
so, max value = 2
table of values look like
note: first differences are changing
note: second differences are constant
e.g.
x
–3
–2
–1
0
1
2
3
y = x2 + 2x + 4
y
7
4
3
4
7
12
19
first
difference
4 – 7 = –2
3 – 4 = –1
4–3=1
7–4=3
12 – 7 = 5
19 – 12 = 7
second
difference
–1 – (–2) = 2
1 – (–1) = 2
3 –1 = 2
5–3=2
7–5=2
15
Quadratic Functions – Intro
quadratic equations
can be in standard form:
y = ax2 + bx + c
where a and b and c are all numbers
but a  0 (can’t be zero), (but b or c can = 0)
so, the obvious possibilities are:
y = ax2 + bx + c
y = ax2 + c
y = ax2
can be in the factored form:
y = a(x – s)(x – t)
y = a(x – s)(x – t)
y = a(x – s)(x – t)
is ok for a quadratic function
is ok for a quadratic function
is ok for a quadratic function
e.g. y = x2 – 7x + 10
e.g. y = x2 + 10
e.g. y = x2
y = a(x – s)(x – t) where a and s and t are all numbers
but a  0 (can’t be zero), (but s or t can = 0)
is ok for a quadratic function
is ok for a quadratic function
is ok for a quadratic function
e.g. y = (x – 2)(x – 5)
e.g. y = (x – 0)(x – 5)
e.g. y = (x – 2)(x – 0)
note: the equation can be in other forms as well…
note: you can change from standard form to factored form by factoring
y = x2 – 7x + 10
y = (x – 2)(x – 5)
what are 2 numbers that multiply to +10 and add up to –7  –2 and –5
note: you can change from factored form to standard form by expanding
y = (x – 2)(x – 5)
use “FOIL”
y = x2 – 5x – 2x + 10
y = x2 – 7x + 10
parabola – “U” shaped curve
“zeros”
vertex (x, y)
axis of symmetry
y-intercept
maximum / minimum value
x-intercepts / “roots” /
graph of a quadratic must be in the form of a: parabola  symmetrical “U” shape
vertex – the point on the graph (parabola) with the greatest y-coordinate value (if the graph opens up), OR
the point on the graph (parabola) with the least y-coordinate value (if the graph opens down)
maximum value – the value of the y-coordinate of the vertex (if the graph opens down)
minimum value – the value of the y-coordinate of the vertex (if the graph opens up)
axis of symmetry – a (vertical) line through the vertex that divides the parabola into 2 (mirror-image) halves
the equation of the axis of symmetry will always be x = some number
(the value of the number is the value of the x-coordinate of the vertex)
x-intercepts (also known as the “roots” or the “zeros” of the quadratic) – where the parabola crosses the x-axis
note: if the parabola opens upwards, and the vertex is above the x-axis, there will be no x-intercepts
note: you can determine the x-intercepts (“zeroes”) (“roots”) of a quadratic:
1) by graphing it (by hand or using a graphing calculator) or 2) by factoring it (see previous page)
note: the average of the values of the x-intercepts is the value of the x-coordinate of the vertex (x, y)
y-intercept – where the parabola crosses the y-axis