Chapter 5- Transformations Of Functions And Quadratic Functions

Front 1

if y=g(x) is a function, and p(x) is the graph of g(x) shifted up vertically by two units, what is p(x) in terms of g(x)?

Back 1

p(x)=g(x)+2

Front 2

If y=u(x) is a function and h(x) is the graph of u(x) shifted left two and down 10, what is the formula for h(x) in terms of u(x)?

Back 2

h(x)=u(x+2)-10

Front 3

If r(x) is a function, then r(x-c) denotes a horizontal shift of r(x) to the _______ and r(x+c) denotes a horizontal shift of r(x) to the _______ by how many units?

Back 3

right
left
c

Front 4

If a function is to be horizontally shifted and also vertically shifted does it matter which order you perform vertical and horizontal shifts?

Back 4

No, you can perform the shifts in any order. The order of operations does matter when you deal with compressions and stretches but not with shifts.

Front 5

If f is an even function than what is algebraically true about f?

Back 5

f(-x)=x

Front 6

How would you prove that
y=(x^3) +5x is an odd function?

Back 6

Need to show that f(-x)=-f(x)
Therefore
f(-x)= (-x)^3 +5(-x)
= -x^3 -5x
and
-f(x) = -1*((x^3) +5x)
= -x^3-5x
so f(-x)=-f(x)

Front 7

How would you show that
y=x^2+2x-1 is an even function?

Back 7

show that f(-x)=f(x)
f(-x)=(-x)^2 +2(-x) -1
= x^2 -2x -1

since this = f(x) the function is even

Front 8

If f is an odd function then it is symmetric with respect to the _______
and obeys which algebraic property?

Back 8

origin
f(-x)=-f(x)

Front 9

If f is an even function then it is symmetric with respect to the _______
and obeys which algebraic property?

Back 9

y axis
f(-x)=f(x)

Front 10

True or false, if a function is symmetric with respect to the origin, then flipping it over the y axis then over the x- axis will give you back the original function?

Back 10

true

Front 11

True or false
If a function is even, you can flip it across the x axis and you will get the same function back

Back 11

false, you must flip it over the y-axis since even functions are symmetric with respect to the y-axis.

Front 12

If f is a function and k is a constant then how does y=kf(x) relate to the graph of y=f(x), if:

1. k>1
2. 0<k<1
3. -1<k<0
4. k<-1

Back 12

1. it is the graph of f(x) stretched vertically by a factor of k
2. it is the graph of f(x) compressed vertically by a factor of k
3. it is the graph of f(x) compressed by a factor of k and then flipped over the x axis
4. It is the graph of f(x) stretched by a factor of k and flipped over the x axis

Front 13

If g(x)=kf(x) then on any interval, what is true about the average rate of change of g compared to the average rate of change of f?

Back 13

average rate of change of g = k times the average rate of change of f

Front 14

how is the graph of y=0.5f(x) related to the graph of f(x)?

Back 14

y = 0.5f(x) is the graph of f(x) compressed vertically by one-half

Front 15

if (2,6) is a point on the graph of f(x) and f(x) is subsequently shifted vertically up by 2, vertically stretched by a factor of 3 and flipped over the x axis, what point is on the graph of f(x) after it has been transformed?

Back 15

vertical shift up by 2 makes (2,6) go to (2,8). vertical stretch by 3 makes (2,8) go to (2,24). Flipping over x axis makes (2, 24) go to (2,-24)

Front 16

How is the graph of f(kx) related to the graph of f(x) if

1. k>1
2. 0<k<1
3. -1<k<0
4. k<-1

Back 16

The graph of f(kx) is the graph of f(x):

1. horizontally compressed by a factor of 1/k
2. horizontally stretched by a factor of 1/k
3. horizontally stretched by a factor of 1/k then flipped over the y-axis
4. horizontally compressed by a factor of 1/k then flipped over the y-axis.

Front 17

if (6,3) is a point on f(x), then what point is on g(x) if g(x)= f(-2x) +4

Back 17

work from inside out:

k=-2 means a horizontal compression of f(x) by a factor of 1/2 and then a flip over the y axis. Thus (6,3) goes to
(3,3) then to (-3,3). The plus 4 shifts the graph vertically by four so (-3,3) goes to (-3,7)

Front 18

True or false. When dealing with horizontal and vertical compressions, stretches and shifts, it does not matter what order you perform the transformations

Back 18

False, when dealing with stretches and compressions in any case, you need to work from the inside of the parenthesis to the outside of the parentheis. For example, to get the the function j(x)= -3f(-2x)-5, if you are given f(x), you would need to first horizontally compress f(x) by a factor of 1/2, then flip it across the y axis, then vertically stretch it by 3, then flip it over the x axis, then shift it down vertically by 5 units.

Front 19

y=e^(x+2) is really just the graph of e^x when it has been.....

Back 19

shifted horizontally to the left by two units.

Front 20

What are the standard and vertex forms of a quadratic function?

Back 20

standard: ax^2 +bx +c
vertex: y=a(x-h)^2 +k

for a, b, c, h, k are constants and a cannot =0

Front 21

What does the point (h,k) stand for in the vertex form of a quadratic function?

Back 21

it is the vertex of the parabola that represents the function.

Front 22

a parabola has axis of symmetry x= what?

Back 22

x=h where (h,k) is the vertex of the parabola

Front 23

if a<0 then the parabola of a function opens __________ and if a>0 the parabola opens_________

Back 23

downward
upward

Front 24

if a<0 then the graph of the quadratic: is concave down or concave up?
has a max or a min?

what about if a>0?

Back 24

concave down, has a max
concave up, has a max

Front 25

to change a quadratic equation from standard form to vertex form one must.......

Back 25

complete the square

example:

2x^2 - 12x + 20
2[x^2-6x+10]
2[(x- ? )^2 + 10 -?]
2[(x-3)^2 + 10 -9]
2[(x-3)^2 +1]
2(x-3)^2 + 2

Front 26

to change a quadratic equation from vertex form to standard form one must.....

Back 26

multiply out the squared term in the vertex form and simplify.

example: y= 2(x-3)^2 -1
=2(x-3)(x-3)-1
= 2[x^2 -6x +9] -1
= 2x^2 -12x +18-1
= 2x^2 -12x +17

Front 27

What are the steps to completing the square for a general function:

y=ax^2 +bx +c

Back 27

1. factor a out of the equation
2. take x and depending on the sign, add or subtract b/2.
3. Square the above term.
4. Subtract (b/2)^2 from the equation
5. simplify and multiply the a factor back into the equation

Front 28

True or false, one can always recognize a quadratic function by whether or not it has a 2 as the highest exponent.

Back 28

True, quadratic equations always take the form y=ax^2 +bx +c

Front 29

y=-f(x) is a reflection of f(x) across which axis?

y=f(-x) is a reflection of f(x) across which axis?

Back 29

x-axis
y-axis

Front 30

True or false,
the graph of every quadratic equation is a parabola

Back 30

True