Download 14 Radicals Packet Part 2

RADICAL FUNCTIONS PART 2: GRAPHING
NAME: _____________________________________________
The Square Root Function: The parent function is
Let’s look at its graph and table of values using our calculator:
The Cube Root Function: The parent function is
Let’s look at its graph and table of values using our calculator:
BLOCK: _______
Functions have the same transformations as the absolute value function y = a|x – h| + k.
Given
if
: Vertically Stretch the graph by a factor of
if
: Vertically Shrink the graph by a factor of
if
: Reflect the graph about the x-axis
(h, k): Translate the graph horizontally h units and vertically k units.
Let c be a positive real number. Let
.
Vertical shift c units upward:
Horizontal shift right c units:
Vertical shift c units downward:
Horizontal shift left c units:
Transformations of Square Root Function:
(h,k): _________
h(x) = 2 x + 1 − 4
(h,k): _________
(h,k): _________
(h,k): _________
With SQUARE ROOT FUNCTIONS when you are completing the table of values…you will have
x-values on ONE side of the initial point (h,k).
Transformations of Cube Root Function:
(h,k): _________
h(x) = −2 x − 4
_________
(h,k): _________
(h,k): _________
f (x) = − 3 x − 2
(h,k):
f (x) = x − 2
(h,k): _________
(h,k): _________
Day 14: Graphing Square & Cube Root Functions
In these notes we will
ANALYZE the graphs of Square Root and Cube Root Functions
Domain Restrictions based on an equation:
1. Dividing by zero is undefined: a denominator can NEVER be equal to zero.
2. The square root of a negative number does not exist . . . we NEVER put a negative
number under a square root (unless we are dealing in complex numbers).
We will look at Case #1 in Unit 7.
Case #2 above: No Negatives Under the Radical Sign!! x
0
Do you have a square root? Do you have a rational power that has a denominator of 2?
If not, then you don’t have to worry about this restriction.
f(x) =
f(x) =
Domain: The set of all real numbers x ≥ 0
Now let’s go back and define our characteristics from Unit 2 with the square root and cube root
function.
The Square Root Function: The parent function is
(h, k): _____
x - Intercepts: ____________
y - intercept: ______________
Domain: _____________________
Range: _______________________
Increasing: ____________________
Decreasing: ___________________
The Cube Root Function: The parent function is
(h, k): _____
x - Intercepts: ____________
y - intercept: ______________
Domain: _____________________
Range: _______________________
Increasing: ____________________
Decreasing: ___________________
Complete the following. Graph without a calculator. Then verify with your calculator and use to find your
intercepts if necessary. Round to the nearest tenth.
(h,k): __________
x-int: ___________
y-int: ___________
Domain: _________
Range: __________
(h,k): __________
x-int: ___________
y-int: ___________
Domain: _________
Range: __________
(h,k): __________
x-int: ___________
y-int: ___________
Domain: _________
Range: __________
(h,k): __________
x-int: ___________
y-int: ___________
Domain: _________
Range: __________
GRAPHING FUNCTIONS SUMMARY
Quadratic Function
Square Root Function
y = x2
Parent Function:
Parent Function:
General Function:
General Function: y = a(x −h)2 +k
y= x
y = a x −h +k
GRAPHING INSTRUCTIONS
GRAPHING INSTRUCTIONS
1.
Plot the vertex: (h, k)
2.
Write the chart.
3.
4.
1
2
3
4
1
4
9
16
1.
Plot the initial point: (h, k)
2.
Write the chart.
3.
Multiply the right side of
the chart by a.
a:
a > 0: graph will point up (inc)
a < 0: graph will point down (dec)
4.
Use the chart to find other points. Make
sure go find each new point from the
vertex.
Square Root Function
Multiply the right side of
the chart by a.
Use the chart to find other points. Make
sure go find each new point from the
vertex.
Cubic Function
Parent Function:
General Function:
a:
1.
2.
3.
y = x3
y = a(x −h)3 +k
a > 0: graph will point up (inc)
a < 0: graph will point down (dec)
GRAPHING INSTRUCTIONS
Plot the key point: (h, k)
Write the chart.
1
1
2
8
Multiply the right side of
the chart by a.
Parent Function:
General Function:
a:
1.
2.
3.
1
4
9
16
1
2
3
4
y =3x
y = a 3 x −h +k
a > 0: graph will point up (inc)
a < 0: graph will point down (dec)
GRAPHING INSTRUCTIONS
Plot the initial point: (h, k)
Write the chart.
1
8
Multiply the right side of
the chart by a.
1
2
4.
Use the chart to find other points. Make
sure go find each new point from the
vertex.
4.
Use the chart to find other points. Make
sure go find each new point from the
vertex.
5.
Make sure that the graph points in the
correct direction.
5.
Make sure that the graph points in the
correct direction.
Absolute Value Function
Parent Function:
y= x
General Function:
a:
h:
Graphs without horizontal or vertical shifts
h is added or subtracted to x
k is added or subtracted to the entire function
y = a x −h +k
a is the slope
+a: Opens up
-a: Opens down
a > 1: Stretches
a < 1: Compresses
x + h: Left
x – h: Right
+k: Up
-k: Down
GRAPHING INSTRUCTIONS
1.
Plot the vertex: (h, k)
2.
Determine if the graph opens up or down.
3.
Use the slope to find other points.
y = x2
Initial Point: (0, 0)
y = x3 +5
Initial Point: (0, 5)
y = −2 x − 3
Initial Point: (0, –3)
y = 3 x −10
Initial Point: (10, 0)
y = 14 x −8
Vertex: (0, –8)
Find the Domain Analytically
Think about the shape!
Function
Domain
Function
1.
y = x2
6.
y = − 4x + 3
2.
y = x3
7.
y = x + 6 −1
3.
y= x
8.
y=3x
4.
y = 2 x +5
9. y = (x + 7)1/2
5.
y = 2x −1
10. f(x) = |x|
Domain