4.1 Notes – Graph
Quadratic Functions
in Standard Form
Quadratic
Function
Standard
Form
Parabola that is a function in the
shape of a U
y = ax2 + bx + c
Max or Min of a parabola x  b
Vertex
Axis of
Symmetry
 b 
 ,?
 2a 
2a
Line that splits the graph vertically
b
x
2a
Transformations of a, b, and c.
y = ax2 + bx + c
a
Positive #
Negative #
Opens up
Opens down
Has minimum
Has maximum
a = 1, normal a = –1, normal
Fraction, wide Fraction, wide
a > 1, skinny a < –1, skinny
x – intercepts
Set y = 0, factor the equation and
solve for x
Two answers
One answer
No Real answers
y = ax2 + bx + c
Shift vertex left or right
b
c
y – intercept
3. Sketch the graph of the quadratic equations. Identify the
vertex, axis of symmetry, the minimum or maximum, the xintercepts, and the y-intercepts. This is the parent function.
yx
b
(0)
x


2a
2 1
0
 0
1
2
3. Sketch the graph of the quadratic equations.
Identify the vertex, axis of symmetry, the
minimum or maximum, the x-intercepts, and the
y-intercepts.
x
2
1
0
–1
–2
yx
2
y  (2)  4
2
y  (1)  1
2
y  (0)  0
2
y  (1)  1
2
y  (2)  4
2
(x,y)
(2, 4)
(1,1)
(0,0)
(–1,1)
(–2, 4)
yx
2
x 0
2
x0
(0,0)
Vertex: ________________
x=0
axis of symmetry: ________
circle one: min or max
y=0
min/max value: _________
0
x – intercept(s): _________
0
y-intercept: _____________
y  x2
(0,0)
Vertex: ________________
x=0
axis of symmetry: ________
circle one: min or max
y=0
min/max value: _________
0
x – intercept(s): _________
0
y-intercept: _____________
(2, 4) (1,1) (0,0) (–1,1) (–2, 4)
1. Put the equation in standard from.
Determine if the graph opens up or down and
if it is normal, skinny, or wide.
1 2
Standard Form: y  x  2 x  11
2
Opens up
wide
1. Put the equation in standard from.
Determine if the graph opens up or down and
if it is normal, skinny, or wide.
y  6x  3x  8
2
Standard Form:
y  3x  6 x  8
2
Opens down
skinny
1. Put the equation in standard from.
Determine if the graph opens up or down and
if it is normal, skinny, or wide.
y  5 x
Standard Form:
2
y  x  5
2
Opens down
normal
2. Identify the vertex. State the max or min.
1 2
y  x  2 x  11
2
b
2
(2)
x

 2

2a
1
1
2 
2
1 2
y  (2)  2(2)  11  2  4  11  13
2
(2, –13)
Vertex: ________________
y = –13
min/max value: _________
2. Identify the vertex. State the max or min.
y  3x  6 x  8
2
b
6
(6)
x

 1

2a 2  3 6
y  3(1)  6(1)  8  3  6  8  11
2
(1, 11)
Vertex: ________________
y = 11
min/max value: _________
2. Identify the vertex. State the max or min.
y  x  5
2
b
0
(0)
x

0

2a
2  1 2
y  (0)  5  0  5  5
2
(0, 5)
Vertex: ________________
y= 5
min/max value: _________
3. Sketch the graph of the quadratic equations. Identify the
vertex, axis of symmetry, the minimum or maximum, the xintercepts, and the y-intercepts.
y  x  9
2
b
0
(0)
x

0

2a
2  1 2
3. Sketch the graph of the quadratic equations.
Identify the vertex, axis of symmetry, the
minimum or maximum, the x-intercepts, and the
y-intercepts.
x
2
1
0
–1
–2
y  x  9
2
y  (2)  9  4  9  5
2
y  (1)  9  1  9  8
2
y  (0)  9  0  9  9
2
(x,y)
(2, 5)
(1, 8)
(0, 9)
(–1, 8)
(–2, 5)
y  x  9
2
(0, 9)
Vertex: ________________
x  9  0
x=0
axis of symmetry: ________
2
( x  9)  0
( x  3)( x  3)  0
2
x + 3 = 0 or x - 3 = 0
x=3
x = -3
circle one: min or max
y=9
min/max value: _________
3
x – intercept(s): _________
9
y-intercept: _____________
y   x2  9
(0, 9)
Vertex: ________________
x=0
axis of symmetry: ________
circle one: min or max
y=9
min/max value: _________
3
x – intercept(s): _________
9
y-intercept: _____________
(2, 5) (1, 8) (0, 9) (–1, 8) (–2, 5)
3. Sketch the graph of the quadratic equations. Identify the
vertex, axis of symmetry, the minimum or maximum, the xintercepts, and the y-intercepts.
1 2
y  x 2
2
b
0
(0)
x

  0
2a
1 1
2 
2
3. Sketch the graph of the quadratic equations.
Identify the vertex, axis of symmetry, the
minimum or maximum, the x-intercepts, and the
y-intercepts.
1 2
y  x 2
2
x
4
2
0
–2
–4
1 2
(4)  2  8  2  6
2
1
y  (2) 2  2  2  2  0
2
1
y  (0) 2  2  0  2  2
2
y
(x,y)
(4, 6)
(2, 0)
(0, –2)
(–2, 0)
(–4, 6)
1 2
y  x 2
2
1 2
x 20
2
1 2
x 2
2
x 4
2
x  2
(0, –2)
Vertex: ________________
x=0
axis of symmetry: ________
circle one: min or max
y = –2
min/max value: _________
2
x – intercept(s): _________
–2
y-intercept: _____________
1 2
y  x 2
2
(0, –2)
Vertex: ________________
x=0
axis of symmetry: ________
circle one: min or max
y = –2
min/max value: _________
2
x – intercept(s): _________
–2
y-intercept: _____________
(4, 6) (2, 0) (0, –2) (–2, 0) (–4, 6)
3. Sketch the graph of the quadratic equations. Identify the
vertex, axis of symmetry, the minimum or maximum, the xintercepts, and the y-intercepts.
y  3x2  6x
b
6
(6)
x

 1

2a 2  3 6
3. Sketch the graph of the quadratic equations.
Identify the vertex, axis of symmetry, the
minimum or maximum, the x-intercepts, and the
y-intercepts.
x
1
0
–1
–2
–3
y  3x  6x
2
(x,y)
y  3(0)2  6(0)  0  0  0
(1, –9)
(0, 0)
y  3(1)2  6(1)  3  6  3
(–1, 3)
y  3(1)2  6(1)  3  6  9
(–2, 0)
(–3, –9)
y  3x  6x
2
3x  6 x  0
2
-3x
-3x
(–1, 3)
Vertex: ________________
x = –1
axis of symmetry: ________
circle one: min or max
-3x(x + 2) = 0
-3x = 0
x=0
or x + 2 = 0
x = -2
y=3
min/max value: _________
0, -2
x – intercept(s): _________
0
y-intercept: _____________
y  3x2  6x
(–1, 3)
Vertex: ________________
x = –1
axis of symmetry: ________
circle one: min or max
y=3
min/max value: _________
0, -2
x – intercept(s): _________
0
y-intercept: _____________
(1, –9) (0, 0) (–1, 3) (–2, 0) (–3, –9)
3. Sketch the graph of the quadratic equations. Identify the
vertex, axis of symmetry, the minimum or maximum, the xintercepts, and the y-intercepts.
y  x  4x  3
2
b
4
(4)
x

 2

2a
2
2 1
3. Sketch the graph of the quadratic equations.
Identify the vertex, axis of symmetry, the
minimum or maximum, the x-intercepts, and the
y-intercepts.
x
0
–1
–2
–3
–4
y  x  4x  3
2
y  (0)2  4(0)  3  0  0  3  3
(x,y)
y  (1)2  4(1)  3  1  4  3  0
(0, 3)
(–1, 0)
y  (2)2  4(2)  3  4  8  3  1
(–2, –1)
(–3, 0)
(–4, 3)
y  x  4x  3
2
x  4x  3  0
2
x
x
x = -2
axis of symmetry: ________
3
circle one: min or max
1
y = –1
min/max value: _________
3x + x
-1, -3
x – intercept(s): _________
(x + 3)(x + 1) = 0
x + 3 = 0 or
x = –3
(–2, –1)
Vertex: ________________
3
y-intercept: _____________
x+1=0
x = –1
y  x  4x  3
2
(–2, –1)
Vertex: ________________
x = -2
axis of symmetry: ________
circle one: min or max
y = –1
min/max value: _________
-1, -3
x – intercept(s): _________
3
y-intercept: _____________
(0, 3) (–1, 0) (–2, –1) (–3, 0) (–4, 3)
3. Sketch the graph of the quadratic equations. Identify the
vertex, axis of symmetry, the minimum or maximum, the xintercepts, and the y-intercepts.
y  2 x2  4 x  2
b
4
(4)
x

 1

2a 2  2  4
3. Sketch the graph of the quadratic equations.
Identify the vertex, axis of symmetry, the
minimum or maximum, the x-intercepts, and the
y-intercepts.
x
3
2
1
0
–1
y  2 x  4 x  2
2
(x,y)
y  2(2)2  4(2)  2  8  8  2  2
(3, –8)
(2, –2)
y  2(1)2  4(1)  2  2  4  2  0
(1, 0)
y  2(3)2  4(3)  2  18 12  2  8
(0, –2)
(–1, –8)
y  2 x  4 x  2
2
2 x  4 x  2  0
(1, 0)
Vertex: ________________
-2
x=1
axis of symmetry: ________
2
-2
-2
-2(x2 – 2x + 1) = 0
x -1
x -1
-x + -x
-2(x – 1)2 = 0
(x – 1)2 = 0
x–1=0
x =1
circle one: min or max
y=0
min/max value: _________
1
x – intercept(s): _________
-2
y-intercept: _____________
y  2 x2  4 x  2
(1, 0)
Vertex: ________________
x=1
axis of symmetry: ________
circle one: min or max
y=0
min/max value: _________
1
x – intercept(s): _________
-2
y-intercept: _____________
(3, –8) (2, –2) (1, 0) (0, –2) (–1, –8)