relations and functionscontent.njctl.org/courses/math/archived-coursesunits...2009/05/13 ·...
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Relations and Functions
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Table of Contents
Midpoint and Distance FormulaCirclesDomain and RangeDiscrete v ContinuousRelations and Functionsy = a f( b(x + c)) + d Transformations
Operations with FunctionsCompositionsInverses
Vertex Form of a Parabola
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Midpoint and DistanceFormula
Return to Tableof Contents
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The Midpoint FormulaGive points A(x1,y1) and B (x2,y2), the point midway between A and B is
Examples: Find the midpoint of the segment with the given endpoints.
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1 Find the midpoint of K(1,8) & L(5,2).
A (2,3)
B (3,5)
C (-2,-3)
D (-3,-5)
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2 Find the midpoint of H(-4 , 8) & L(6, 10).
A (2,8)
B (-5,-2)
C (-2,-8)
D (5,2)
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3 Given the midpoint of a segment is (4 , 9) and one endpoint is (-3 , 10), find the other midpoint.
A (-10 , 8)
B (11 , 8)
C (-10 , 11)
D (.5 , 9.5)
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Example: If the distance between (3, -2) and (8, y) is 6, find the possible values of y.
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4 What is the distance between (2, 4) and (-1, 8)?
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5 What is the distance between (0, 7) and (5, -5)?
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6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x.
A -7
B -5
C -3
D -1
E 0
F 1
G 3
H 5
I 7
J 9
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Circles
Return to Tableof Contents
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7 Write the equation of the circle with center (5 , 2) and radius 6
A
B
C
D
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8 Write the equation of the circle with center (-5 , 0) and radius 7
A
B
C
D
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9 Write the equation of the circle with center (-2 , 1) and radius
A
B
C
D
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10 What is the center and radius of the following equation?
A
B
C
D
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12 What is the center and radius of the following equation?
A
B
C
D
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13 What is eccentricity of a circle?
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Write the equation of the circle in standard form that meets the following criteria:
Complete the square for the x's
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14 What is the equation of the circle that has a diameter with endpoints (0 , 0) and (16 , 12)?
A
B
C
D
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15 What is the equation of the circle with center (-3 , 5) and contains point (1, 3)?
A
B
C
D
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16 What is the equation of the circle with center (7 , -3) and tangent to the x-axis?
A
B
C
D
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Domain and Range
Return toTable of Contents
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Domain is the values of x that work for a given relation or equation.
What is the domain for each of the following?
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Domain is the values of x that work for a given relation or equation.
What is the domain for each of the following?
234
7-3 8
-2 3-5
412
589
x y x y x y
{2, 3, 4} {1, 2}{-2, 3, -5}
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Domain is the values of x that work for a given relation or equation.
What is the domain for each of the following?
X Y
1 2
2 3
3 2
4 3
5 2
X Y
1 3
2 4
5 -5
3 9
4 7
X Y
-3 4
-1 5
0 8
-1 9
3 11
{1, 2, 3, 4, 5} {1, 2, 3, 4, 5} {-3, -1, 0, 3}
Slide 38 / 233
Domain is the values of x that work for a given relation or equation.
What is the domain for each of the following?
{-2, 0, 2, 3} {reals} {-3 < x < -1}
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Domain is the values of x that work for a given relation or equation.
What is the domain for each of the following?
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19 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
{(3,1), (2,-1), (1,1)}
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20 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
-103
-2-1 0
x y
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21 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
X Y
-2 3
0 2
-1 -1
3 2
-2 0
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22 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
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23 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
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24 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
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25 What is the domain of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H x<0
I x>0
J Reals
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Range is the values of y that work for a given relation or equation.
What is the range for each of the following?
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234
7-3 8
-2 3-5
412
589
x y x y x y
{-3, 7, 8} {5, 8, 9}{4}
Range is the values of y that work for a given relation or equation.
What is the range for each of the following?
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X Y
1 2
2 3
3 2
4 3
5 2
X Y
1 3
2 4
5 -5
3 9
4 7
X Y
-3 4
-1 5
0 8
-1 9
3 11
{-5, 3, 4, 7, 9} {2, 3} {4, 5, 8, 9, 11}
Range is the values of y that work for a given relation or equation.
What is the range for each of the following?
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{-3, -2, -1, 2, 3} {2} {-3 < y < 4}
Range is the values of y that work for a given relation or equation.
What is the range for each of the following?
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Range is the values of y that work for a given relation or equation.
What is the range for each of the following?
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26 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
{(3,1), (2,-1), (1,1)}
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27 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
-103
-2-1 0
x y
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28 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
X Y
-2 3
0 2
-1 -1
3 2
-2 0
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29 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
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30 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
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31 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
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32 What is the range of the following?
A -3
B -2
C -1
D 0
E 1
F 2
G 3
H y<0
I y>0
J Reals
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Discrete v Continuous
Return toTable of Contents
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A relation is discrete if it is made up separate points.
For example, you go to a bakery to buy donuts. How many can you buy? 0, 1, 2, 3, . . . These are separate values. 1.2, 1.375, 1.5899 do not have meaning.
A relation is continuous if the points are not separate.
For example, the repairman says he will be to your home between 1 and 5. What time could he show up? 1, 2, 3, 4, 5. Do the values between 1 and 2 have meaning?
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Are the following relations discrete or continuous?
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33 Is the given relation discrete or continuous?
A Discrete
B Continuous
{(3,1), (2,-1), (1,1)}
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34 Is the given relation discrete or continuous?
A Discrete
B Continuous
-103
-2-1 0
x y
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35 Is the given relation discrete or continuous?
A Discrete
B Continuous
X Y
-2 3
0 2
-1 -1
3 2
-2 0
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36 Is the given relation discrete or continuous?
A Discrete
B Continuous
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37 Is the given relation discrete or continuous?
A Discrete
B Continuous
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38 Is the given relation discrete or continuous?
A Discrete
B Continuous
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39 Is the given relation discrete or continuous?
A Discrete
B Continuous
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Relations and Functions
Return toTable of Contents
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A function is a relation that has each value in the domain has exactly one value in the range. In other words, x does not repeat.
Is each of the following relations a function?
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234
7-3 8
-2 3-5
412
589
x y x y x y
A function is a relation that has each value in the domain has exactly one value in the range. In other words, x does not repeat.
Is each of the following relations a function?
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X Y
1 2
2 3
3 2
4 3
5 2
X Y
1 3
2 4
5 -5
3 9
4 7
X Y
-3 4
-1 5
0 8
-1 9
3 11
A function is a relation that has each value in the domain has exactly one value in the range. In other words, x does not repeat.
Is each of the following relations a function?
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On a graph, a function does not have a point above another point. This is called the Vertical Line Test.
Is each of the following relations a function?
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An equation is a function only if when an x is substituted in there is only 1 y-value.
Is each of the following relations a function?
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40 Is the following relation a function?
Yes
No
{(3,1), (2,-1), (1,1)}
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41 Is the following relation a function?
Yes
No
-103
-2-1 0
x y
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42 Is the following relation a function?
Yes
No
X Y
-2 3
0 2
-1 -1
3 2
-2 0
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43 Is the following relation a function?
Yes
No
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44 Is the following relation a function?
Yes
No
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45 Is the following relation a function?
Yes
No
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46 Is the following relation a function?
Yes
No
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y = a f( b(x + c)) + d
Return toTable ofContents
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Vertical Shifts
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Vertical Shifts occur when a constant is added to a function.
The parent function y = f(x) is slid:upward if c > 0
downward if c < 0
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Let the graph of f(x) be
Graph y = f(x) + 2
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Let the graph of f(x) be
Graph y = f(x) - 3
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Let the graph of f(x) be
Graph y = f(x) + 1
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Let the graph of f(x) be
Graph y = f(x) - 1
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Consider the graph y = x2 and the rules for vertical shifts,
Graph
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Consider the graph y = x2 and the rules for vertical shifts,
Graph
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Consider the graph y = |x| and the rules for vertical shifts,
Graph
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Consider the graph y = |x| and the rules for vertical shifts,
Graph
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Consider the graph y = #x and the rules for vertical shifts,
Graph
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47 Given the graph of h(x), which of the following graphs is y = h(x) +1 ?
A B
C D
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48 Given the graph of h(x), which of the following graphs is y = h(x) - 1 ?
A B
C D
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Horizontal Shifts
Slide 100 / 233
Horizontal Shifts occur when a constant is added to a function.
The parent function y = f(x) is slid:left if c > 0
right if c < 0
Notice the direction is opposite the sign of c.
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Let the graph of f(x) be
Graph y = f(x + 2)
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Let the graph of f(x) be
Graph y = f(x - 3)
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Let the graph of f(x) be
Graph y = f(x + 1)
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Let the graph of f(x) be
Graph y = f(x - 1)
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Consider the graph y = x2 and the rules for horizontal shifts,
Graph
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Consider the graph y = x2 and the rules for horizontal shifts,
Graph
Slide 107 / 233
Consider the graph y = |x| and the rules for horizontal shifts,
Graph
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Consider the graph y = |x| and the rules for horizontal shifts,
Graph
Slide 109 / 233
Consider the graph y = #x and the rules for horizontal shifts,
Graph
Slide 110 / 233
Consider the graph y = #x and the rules for horizontal shifts,
Graph
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49 Given the graph of h(x), which of the following graphs is y = h(x + 1) ?
A B
C D
Slide 112 / 233
50 Given the graph of h(x), which of the following graphs is y = h(x - 1)?
A B
C D
Slide 113 / 233
Reflections
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Reflection over the x-axis:
Reflection over the y-axis:
Slide 115 / 233
Let the graph of f(x) be
Graph y = f(-x)
Slide 116 / 233
Let the graph of f(x) be
Graph y = - f(x)
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Consider the graph y = x2 and the rules for reflections,
Graph
Slide 118 / 233
Slide 119 / 233
Consider the graph y = |x| and the rules for reflections,
Graph
Slide 120 / 233
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Slide 121 / 233
Consider the graph y = #x and the rules for reflections,
Graph
Slide 122 / 233
Consider the graph y = #x and the rules for reflections,
Graph
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51 Given the graph of h(x), which of the following graphs is y = -h(x) ?
A B
C D
Slide 124 / 233
52 Given the graph of h(x), which of the following graphs is y = h(-x) ?
A B
C D
Slide 125 / 233
53 Given the graph of h(x), which of the following graphs is y = -h(-x) ?
A B
C D
Slide 126 / 233
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Vertical Stretch & Shrink
Slide 127 / 233
Vertical Stretches and Shrinks occur when a constant is multiplied to a
function.
The parent function y = f(x) is :streched if |c| > 1
shrunk if 0 < |c| < 1
Stretches and shrinks are the first transformation that do not yield
congruent figures.
Slide 128 / 233
Let the graph of f(x) be
Graph y = 2f(x)
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Let the graph of f(x) be
Graph y =(1/3 )f(x)
Slide 130 / 233
Let the graph of f(x) be
Graph y = 3f(x)
Slide 131 / 233
Let the graph of f(x) be
Graph y = (1/2)f(x)
Slide 132 / 233
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Consider the graph y = x2 and the rules for stretches and shrinks,
Graph
Slide 133 / 233
Consider the graph y = x2 and the rules for stretches and shrinks,
Graph
Slide 134 / 233
Consider the graph y = |x| and the rules for stretches and shrinks,
Graph
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Consider the graph y = |x| and the rules for stretches and shrinks,
Graph
Slide 136 / 233
Consider the graph y = #x and the rules for stretches and shrinks,
Graph
Slide 137 / 233
Consider the graph y = #x and the rules for stretches and shrinks,
Graph
Slide 138 / 233
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54 Given the graph of h(x), which of the following graphs is y = 2h(x)?
A B
C D
Slide 139 / 233
55 Given the graph of h(x), which of the following graphs is y =1/2 h(x)?
A B
C D
Slide 140 / 233
Horizontal Stretch & Shrink
Slide 141 / 233
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Horizontal Stretches and Shrinks occur when a constant is multiplied to
x in a function.
The parent function y = f(x) is:shrunk if |c| > 1
stretched if 0 < |c| < 0
Horizontal stretches and shrinks will also impact the vertical stretches and shrinks.
Slide 142 / 233
Let the graph of f(x) be
Graph y = f(2x)
Slide 143 / 233
Let the graph of f(x) be
Graph y = f(3x)
Slide 144 / 233
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Let the graph of f(x) be
Graph y = f(.5x)
Slide 145 / 233
Let the graph of f(x) be
Graph y = f(.75x)
Slide 146 / 233
Consider the graph y = x2 and the rules for stretches and shrinks,
Graph
Slide 147 / 233
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Consider the graph y = x2 and the rules for stretches and shrinks,
Graph
Slide 148 / 233
Consider the graph y = |x| and the rules for stretches and shrinks,
Graph
Slide 149 / 233
Consider the graph y = |x| and the rules for stretches and shrinks,
Graph
Slide 150 / 233
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Consider the graph y = #x and the rules for stretches and shrinks,
Graph
Slide 151 / 233
Consider the graph y = #x and the rules for stretches and shrinks,
Graph
Slide 152 / 233
56 Given the graph of h(x), which of the following graphs is y = h(2x)?
A B
C D
Slide 153 / 233
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57 Given the graph of h(x), which of the following graphs is y = h(1/2x)?
A B
C D
Slide 154 / 233
Combining Transformations
Slide 155 / 233
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Let the graph of f(x) be
Graph y = 2f(.5x+1) - 2
Slide 157 / 233
Let the graph of f(x) be
Graph y =(-1/3 )f(2x + 1) + 2
Slide 158 / 233
Let the graph of f(x) be
Graph y = 3f(-.5x - 2) + 1
Slide 159 / 233
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Let the graph of f(x) be
Graph y = (-1/2)f(-x + 2) +1
Slide 160 / 233
Consider the graph y = x2 and the rules for stretches and shrinks,
Graph
Slide 161 / 233
Consider the graph y = |x| and the rules for stretches and shrinks,
Graph
Slide 162 / 233
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Consider the graph y = #x and the rules for stretches and shrinks,
Graph
Slide 163 / 233
58 Given the graph of h(x), which of the following graphs is y = 2h(-x+1) - 3?
A B
C D
Slide 164 / 233
59 Given the graph of h(x), which of the following graphs is y = -0.5h(2x - 1) + 2?
A B
C D
Slide 165 / 233
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Vertex Form of a
ParabolaReturn toTable ofContents
Slide 166 / 233
Vertex FormA quadratic can be written in vertex form:
Vertex form shows the location of the vertex (h , k).The a still tells the direction of the openness.
Example: Find the vertex, direction of openness and the axis of symmetry of:
Vertex: (6 , 4)
Direction of openness is up
Axis of Symmetry: x = 6
Slide 167 / 233
Find the vertex, direction of openness and the axis of symmetry of:
Slide 168 / 233
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60 What is the vertex of
A
B
C
D
Slide 169 / 233
61 What is direction of openness of
A up
B down
C left
D right
Slide 170 / 233
62 The axis of symmetry of is x =
Slide 171 / 233
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Slide 172 / 233
64 What is direction of openness of
A up
B down
C left
D right
Slide 173 / 233
65 The axis of symmetry of is x =
Slide 174 / 233
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Slide 175 / 233
67 What is direction of openness of
A up
B down
C left
D right
Slide 176 / 233
68 The axis of symmetry of is x =
Slide 177 / 233
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Slide 178 / 233
Slide 179 / 233
Slide 180 / 233
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Converting from Standard Form to Vertex FormTo convert, use completing the square.
Step 1 - Write the equation in the form y = x2 + bx +__+ c - __
Step 2 - Find (b ÷ 2)2
Step 3 - Complete the square by adding (b ÷ 2)2 to the first blank and subtracting it in the second blank.
Step 4 - Factor the perfect square trinomial.
Slide 181 / 233
Let's look at an example to solve:
y= x2 + 14x +__+ 15 -__ Step 1
(14 ÷ 2)2 = 49 Step 2 - Find (b÷2)2
y= x2 + 14x + 49 + 15 - 49 Step 3 - Complete the Square
y= (x + 7)2 - 34 Step 4 - Factor and simplify
y= x2 + 14x + 15
Slide 182 / 233
Let's look at an example to solve:
y= x2 - 12x +__- 16 -__ Step 1
(12 ÷ 2)2 = 36 Step 2 - Find (b÷2)2
y= x2 - 12x + 36 - 16 - 36 Step 3 - Complete the Square
y= (x + 6)2 - 52 Step 4 - Factor and simplify
y= x2 - 12x - 16
Slide 183 / 233
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72 What is the vertex form of:
A
B
C
D
Slide 184 / 233
73 What is the vertex form of:
A
B
C
D
Slide 185 / 233
74 What is the vertex form of:
A
B
C
D
Slide 186 / 233
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Converting from Standard Form to Vertex FormWhat if "a" does not equal 1?
Step 1 - Write the equation in the form y = ax2 + bx +__+ c - __
Step 2 - Factor: y = a(x2 + (b/a)x +__)+ c - __
Step 3 - Find (b/a ÷ 2)2
Step 4 - Complete the square by adding (b/a ÷ 2)2 to the first blank and subtracting a(b/a ÷ 2)2 in the second blank.
Step 5 - Factor the perfect square trinomial.
Slide 187 / 233
Let's look at an example to solve:
y= 3x2 + 12x +__- 10 -__ Step 1
y= 3(x2 + 4x + __) - 10 - __ Step 2 - Factor out "a"
(4 ÷ 2)2 = 4 Step 3 - Find (b÷2)2
y= 3(x2 + 4x + 4) - 10 - 12 Step 4 - Complete the Square
y= 3(x + 2)2 - 22 Step 5 - Factor and simplify
y= 3x2 + 12x - 10
Slide 188 / 233
Let's look at an example to solve:
y= -4x2 - 32x +__+ 45 -__ Step 1
y= -4(x2 + 8x + __) + 45 - __ Step 2 - Factor out "a"
(8 ÷ 2)2 = 16 Step 3 - Find (b÷2)2
y= -4(x2 - 8x + 16) + 45 - -64 Step 4 - Complete the Square
y= -4(x - 4)2 + 109 Step 5 - Factor and simplify
y= -4x2 - 32x + 45
Slide 189 / 233
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Slide 190 / 233
Slide 191 / 233
Slide 192 / 233
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Operations with Functions
Return toTable ofContents
Slide 193 / 233
Functions can be combined to make other functions.Given: andFind if:
Slide 194 / 233
78 Given and , find h(x) if
A
B
C
D
Slide 195 / 233
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79 Given and , find h(x) if
A
B
C
D
Slide 196 / 233
80 Given and , find h(x) if
A
B
C
D
Slide 197 / 233
81 Given and , find h(x) if
A
B
C
D
Slide 198 / 233
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Compositions
Return toTable ofContents
Slide 199 / 233
Composite functions are when functions is substituted into a second function.
There are 2 ways of writing a composite:
Method #1:
Method #2:
Slide 200 / 233
To simplify a composite of functions, substitute into the "inner" function, simplify, and then substitute that value in for the variable in the "outer" function.Examples: and
Slide 201 / 233
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82 Find the value of
A
B
C
D
Slide 202 / 233
83 Find the value of
A
B
C
D
Slide 203 / 233
84 Find the value of
A
B
C
D
Slide 204 / 233
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85 Find the value of
A
B
C
D
Slide 205 / 233
86 Find the value of
A
B
C
D
Slide 206 / 233
Inverses
Return toTable ofContents
Slide 207 / 233
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Slide 208 / 233
87 What is ?
A undefined
B 0
C 2
D infinite solutions
Slide 209 / 233
88 What is ?
A undefined
B x
C 2x
D infinite solutions
Slide 210 / 233
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The graph of an inverse is the reflection of the function over y=x.
Can you make a conjecture about the x- and y-values of the function and it's inverse?
Slide 211 / 233
Draw the inverse of the given function.
Is the inverse a function? Why or why not?
Slide 212 / 233
Draw the inverse of the given function.
Is the inverse a function? Why or why not?
Slide 213 / 233
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To determine if an inverse is going to be a function, use the Horizontal Line Test. That is, "Does every horizontal line drawn touch the function no more than once?" It only takes one horizontal line to touch more than once for the function to fail the horizontal line test and there for it's inverse would not be a function.
Slide 214 / 233
89 Will the inverse of the given function be a function?
Yes
No
Slide 215 / 233
90 Which graph is the inverse of the given function?
A
B
C D
Slide 216 / 233
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91 Which graph is the inverse of the given function?
A
B
C D
Slide 217 / 233
92 Which graph is the inverse of the given function?
A
B
C D
Slide 218 / 233
93 Will the inverse of f(x)= 2x2 - 23x +4 be a function?
Yes
No
Slide 219 / 233
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The inverse of a function is the reflection over y=x which results in the swapping of x- and y- values.
Example: Find the inverse of f(x)= {(1,2),(3,5),(-7,6)}
f-1(x)={(2,1),(5,3),(6,-7)
Example: Find the inverse of X Y
3 2
4 4
5 -5
6 7
X Y
2 3
4 4
-5 5
7 6
Slide 220 / 233
94 What is the inverse of {(1,4), (5,3),(2,-1)}?
A {(4,1), (3,5), (2,-1)}
B {(-1,-4), (-5,-3), (-2,1)}
C {(4,1), (3,5),(-1,2)}
D {(-4,-1), (-3,-5), (1,-2 )}
Slide 221 / 233
95 If the inverse of a function is {(1,0), (3,3),(-4,-5)}, what was the original function?A {(0,1), (3,3), (-5,-4)}
B {(-1,-4), (-5,-3), (-2,1)}
C {(0,-1), (-3,-3),(4,5)}
D {(0,1), (3,3), (-4,-5)}
Slide 222 / 233
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96 What is the inverse of:
A B C DX Y
4 3
0 1
3 -2
7 4
X Y
4 3
1 0
2 3
4 7
X Y
4 3
0 0
-2 3
7 4
X Y
-3 -4
-1 0
-2 -3
-4 -7
X Y
3 4
1 0
-2 3
4 7
Slide 223 / 233
97 Will the inverse of be a function?
Yes No
X Y
1 2
2 3
3 4
5 2
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Knowing that the inverse of a function switches x- and y-values, this fact can be used when given an equation.
Given:
Swap x and y's:Solve for y:
So the inverse of is
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Find the inverse of the following functions.
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98 Which of the following choices is the inverse of f(x) if
A
B
C
D
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99 Which of the following choices is the inverse of f(x) if
A
B
C
D
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100 Which of the following choices is the inverse of f(x) if
A
B
C
D
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101 Find the inverse of
A
B
C
D
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102 Find the inverse of
A
B
C
D
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103 What is the inverse of y = 3?
A y = 1/3
B x = 3
C x = 1/3
D not possible
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104 Find the inverse of
A
B
C
D
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