Glossary of chapter 5- transformations of functions and quadratic functions
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- 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)?
- 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)?
- 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?
- If a function is to be horizontally shifted and also vertically shifted does it matter which order you perform vertical and horizontal shifts?
- 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.
- If f is an even function than what is algebraically true about f?
- How would you prove that
y=(x^3) +5x is an odd function?
- Need to show that f(-x)=-f(x)
f(-x)= (-x)^3 +5(-x)
= -x^3 -5x
-f(x) = -1*((x^3) +5x)
- How would you show that
y=x^2+2x-1 is an even function?
- 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
- If f is an odd function then it is symmetric with respect to the _______
and obeys which algebraic property?
- If f is an even function then it is symmetric with respect to the _______
and obeys which algebraic property?
- y axis
- 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?
- True or false
If a function is even, you can flip it across the x axis and you will get the same function back
- false, you must flip it over the y-axis since even functions are symmetric with respect to the y-axis.
- 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. 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
- 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?
- average rate of change of g = k times the average rate of change of f
- how is the graph of y=0.5f(x) related to the graph of f(x)?
- y = 0.5f(x) is the graph of f(x) compressed vertically by one-half
- 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?
- 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)
- How is the graph of f(kx) related to the graph of f(x) if
- 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.
- if (6,3) is a point on f(x), then what point is on g(x) if g(x)= f(-2x) +4
- 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)
- True or false. When dealing with horizontal and vertical compressions, stretches and shifts, it does not matter what order you perform the transformations
- 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.
- y=e^(x+2) is really just the graph of e^x when it has been.....
- shifted horizontally to the left by two units.
- What are the standard and vertex forms of a quadratic function?
- standard: ax^2 +bx +c
vertex: y=a(x-h)^2 +k
for a, b, c, h, k are constants and a cannot =0
- What does the point (h,k) stand for in the vertex form of a quadratic function?
- it is the vertex of the parabola that represents the function.
- a parabola has axis of symmetry x= what?
- x=h where (h,k) is the vertex of the parabola
- if a<0 then the parabola of a function opens __________ and if a>0 the parabola opens_________
- 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?
- concave down, has a max
concave up, has a max
- to change a quadratic equation from standard form to vertex form one must.......
- complete the square
2x^2 - 12x + 20
2[(x- ? )^2 + 10 -?]
2[(x-3)^2 + 10 -9]
2(x-3)^2 + 2
- to change a quadratic equation from vertex form to standard form one must.....
- multiply out the squared term in the vertex form and simplify.
example: y= 2(x-3)^2 -1
= 2[x^2 -6x +9] -1
= 2x^2 -12x +18-1
= 2x^2 -12x +17
- What are the steps to completing the square for a general function:
y=ax^2 +bx +c
- 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
- True or false, one can always recognize a quadratic function by whether or not it has a 2 as the highest exponent.
- True, quadratic equations always take the form y=ax^2 +bx +c
- y=-f(x) is a reflection of f(x) across which axis?
y=f(-x) is a reflection of f(x) across which axis?
- True or false,
the graph of every quadratic equation is a parabola
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