linear line
almost all linear relationships are
function
EXCEPT vertical lines (no y-intercept, all
the same x-coordinates, undefined
slope, no change in x - no run, x = #)
slope, y-intercept,
direct variation ( line thru the origin)
slope-intercept form: y = mx + b
standard form: ax + by = c
point slope: y - y1 = m(x - x1)
degree of 1
constant rate of change
quadratics have a degree of 2
graphs are parabola (u shape)
standard form: y = ax2 + bx + c
a is the coefficient of the quadratic
term
b is the coefficient of the linear term
c is the constant
10.1 Exploring Quadratic Graphs
Lets get out our graphing calculators . . .
• start with the quadratic function x2
we can write y = x2 OR f(x) = x2
y=x
m=1
passes thru origin
b=0
uphill
What happens when we change the coefficient of the quadratic (a) . . .
• like 2x2
we can write y = 2x2 OR f(x) = 2x2
• Which function above represents y = x2 and which function
represents y = 2x2?
• What would happen if we changed the quadratic coefficient to 10, to
20, to 100? (Test your theories using your graphing calculator.)
• What would happen if we changed the quadratic coefficient to -1, to -2, to
-10? (Test your theories using your graphing calculator.)
y = -x2
• What would happen if we changed the quadratic coefficient to 0.5, to 0.1,
to 0.01? (Test your theories using your graphing calculator.)
y = 0.5x2
• Which graph represents y = x2 and y = 0.5x2?
• How does the coefficient of the quadratic term affect the graph?
If a > 1, the graph becomes steeper and more narrow
If 0 < a < 1, the graph becomes wider
If -1 < a < 0, the graph reflects over the x-axis and becomes more narrow
If a < -1, the graph reflects over the y-axis and becomes wider
Identify the vertex of each graph. Determine whether the vertex is a
minimum or maximum. State the axis of symmetry.
Order each group of quadratic function from widest to narrowest graphs.
Make a table:
Graph the points:
You try:
f(x) = 4x2
x
4x2
f(x)
0
4(0)(0)
0
1
4(1)(1)
4
2
4(2)(2)
16
You try:
f(x) = -3x2
x
-3x2
f(x)
0
-3(0)(0)
0
1
-3(1)(1)
-3
2
-3(2)(2)
-12
Notice all of the quadratic function we have looked at have a constant of
_______
• What would happen if we changed the constant? (Test your theories
using your graphing calculator.)
• Let's try y = x2 + 1
y = x2+1
What happened? What would happen if we made the constant 5, or 8?
(Test your theories using your graphing calculator.)
• What would happen if we made the constant negative? (Test your
theories using your graphing calculator.)
• Let's try y = x2 - 4
y = x2 - 4
What happened? What would happen if we made the constant -5, or -8?
(Test your theories using your graphing calculator.)
Let's graph a few:
f(x) = x2-3
x
x2 - 3
f(x)
0
(0)(0) - 3
-3
1
(1)(1) - 3
-2
2
(2)(2) - 3
1
Let's graph a few:
f(x) = x2+ 7
x
x2 + 7
f(x)
Let's graph a few:
f(x) = -2x2+ 3
x
-2x2 + 3
f(x)
0
-2(0)(0) + 3
3
1
-2(1)(1) + 3
1
2
-2(2)(2) + 3
-5
Let's graph a few:
f(x) = 0.25x2-3
OR
x
0.25x2 - 3
f(x)
x
0
0
1
4
2
8
0.25x2 - 3
f(x)
Why did I use 0, 4, and 8?