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115 Cards in this Set
- Front
- Back
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What is a function?
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A relation in which for every input, there is exactly one output.
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In functions, causality...
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... isn't implied
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Is a circle a function?
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No
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Is a horizontal line a function?
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Yes
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Is a vertical line a function?
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No
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Is any line with an existing slope a function?
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Yes
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Is a parabola a function?
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Yes
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Is a "sideways" parabola a function?
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No
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Is a point a function?
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Yes
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Is a semi-circle a function?
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Yes
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How can you tell if something is a function?
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With the vertical line test- imagine a vertical line moving across the function. If it touches more than one point at any time, then it isn't a function.
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What are the four ways to represent a function?
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In a(n):~Equation~Graph~Table~Verbal Statement
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What is a continuous function?
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A function that can be traced along to end without lifting the pencil
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What is a discontinuous function?
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A function that can't be traced along to end without lifting the pencil (with a hole, usually- indicating a rational function with a special denominator condition)
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When adding/subtracting functions, you...
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put both in parentheses and add like terms (x^2, x, constants, etc.)
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When multiplying functions, you...
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FOIL
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A^m * A^n=
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A^(m+n)
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A^m/A^n=
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A^(m-n)
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(A^m)^n=
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A^(m*n)
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A^-m=
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1/(A^m)
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1/(A^-m)=
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A^m
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When operating on exponents...
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the bases must be equal
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What is a composition function?
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A situation in which a function's output is the input of another function (so f(g(x)) is where you find x for g, then use that as x for f)
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A^(m/n)=
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(^n√p)^m ((the nth root of p) to the power of m)
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What/How- horizontal (x) intercept?
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Where a line crosses the x-axis (set y to 0, solve)
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What/How- vertical (y) intercept?
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Where a line crosses the y-axis (set x to 0, solve)
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What is standard form (lines)?
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A way to write the equation of a line- Ax+By=c (Where B and c are integers, and A is a positive integer)
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Exponential function formula?
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y=ab^x
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Domain of an exponential function
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( -oo , oo ) or ℝ
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Range of an exponential function
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y>0 or ℝ+
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Initial value of an exponential function
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a
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Vertical intercept of an exponential function
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(0,a)
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Horizontal intercept of an exponential function
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DNE
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Domain of a quadratic function
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( -oo, oo ) or ℝ
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Vertex (point where direction changes) of a quadratic function
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(h,k) where h is the x-value and k is the y-value
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Range of a quadratic function
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Changes based on concavity- If concave up, the range is y>k (anything greater than the y-value of the vertex coordinate)If concave down, the range is y
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What is the equation for the line of symmetry in a quadratic function?
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x= -b/2a
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Vertical intercept of a quadratic function
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(0,c)
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How do you find h (vertex's x-value) for a quadratic?
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-b/2a
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How do you find k (vertex's y-value) for a quadratic?
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plug in h (vertex's x-value) and solve
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(Quadratics) What happens when a>0?
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The parabola concaves up
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(Quadratics) What happens when a<0?
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The parabola concaves down
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Domain of a radical function
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px>=0 (px is greater than or equal to 0)
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Range of a radical function (odd root)?
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(For example, cube or 5th root), the range is ℝ
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Range of a radical function (even root)?
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(For example, square or 4th root), the value inside the radical must be greater than or equal to 0 (otherwise it is undefined)
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Vertical intercept of a radical function
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x=0
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Horizontal intercept of a radical function
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0=√px
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Domain of a rational function
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All real numbers except the value where the denominator equals 0 (for instance, in f(x)= 3/x-6 the domain is anything except x=6, because 6-6=0)
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Growth Factor:
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Growth Rate + 1 (so if growth rate is 0.25, growth factor is 1.25)*shown in the equation as b
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Growth rate:
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Essentially, b-1. It is the rate at which the function increases (or decreases, if it is a decay rate)
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How can you tell if it is a growth or decay rate?
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If b>1, it is a growth rateIf 0
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If a logarithm doesn't have a given "b" value, assume it is...
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10
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Domain of a logarithm?
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ℝ+
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Range of a logarithm?
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ℝ
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b^0=
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1
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b^1=
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b
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1/b=
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b^-1
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1/b^n=
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b^-n
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√b=
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b^1/2
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n√b (nth root of b)
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b^1/n
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(Property?)
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b^y=x
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If the x value in a log is 1, y=
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0
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(Property?)
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n
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If the x value in a log is 0, y=
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DNE
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i=
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√-1
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i^2=
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-1
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i^3=
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-i
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i^4=
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1
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i^5=
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i
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(rational functions) If the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is...
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DNE
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(rational functions) If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is...
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at the ratio of the the lead coefficients (i.e. 3x^2/x^2 lead coefficients are 3 and 1, ratio 3:1, horizontal asymptote @ y=3)
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(rational functions) If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is...
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at y=0
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Define an isosceles triangle |
A triangle with two congruent sides and congruent base angles |
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Area of a square |
s^2 (s is sidelength) |
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Perimeter of a square |
4s (s is sidelength) |
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Area of a rectangle |
lw (l is length, w is width) |
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Perimeter of a rectangle |
2l+2w (l is length, w is width) |
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Area of a triangle |
1/2 bh (b is base, h is height) |
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Perimeter of a triangle |
s1+s2+s3 (s1 is sidelength 1, etc.) |
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Area of a rhombus |
bh (b is base, h is height) |
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Perimeter of a Rhombus |
4s (s is sidelength) |
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Area of a trapezoid |
1/2 h(b1+b2) (h is height, b1 is base 1, b2 is base 2) |
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Perimeter of a trapezoid |
s1+s2+s3+s4 (s1 is sidelength 1, etc.) |
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Area of a circle |
πr^2 (r is radius) |
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Circumference of a circle |
2πr or πD (r is radius, D is diameter) |
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In a 30-60-90 triangle... |
a is the shortest leg, the other leg is a√3, and the hypotenuse is 2a |
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In a 45-45-90 triangle... |
the legs are a, the hypotenuse is a√2 |
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What is the midpoint formula? |
(((x1+x2)/2),((y1+y2)/2)) |
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If the length scale factor is a:b, what is the area scale factor? |
a^2:b^2 |
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If the length scale factor is a:b, what is the volume scale factor? |
a^3:b^3 |
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Volume of a sphere- |
4/3πr^3 |
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Surface area of a sphere- |
4πr^2 |
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Volume of a cube |
s^3 |
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Surface area of a cube- |
6s^2 |
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Volume of a rectangular prism- |
lwh |
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Surface area of a rectangular prism- |
2(wl+hw+lh)
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Volume of a cylinder- |
πr^2h
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Surface area of a cylinder- |
2πr^2+2πrh |
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Volume of a cone- |
πr^3 (h/3) |
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Surface area of a cone- |
πr(r+√r^2+h^2) |
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Volume of a rectangular pyramid- |
(lwh)/3 |
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Surface area of a rectangular pyramid- |
(lw)+(l√((w/2)^2+h^2))+(w√((1/2)^2+h^2) |
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Sine |
Opposite/Hypotenuse |
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Cosine |
Adjacent/Hypotenuse |
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Tangent |
Opposite/Adjacent |
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Secant |
Hypotenuse/Adjacent |
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Cosecant |
Hypotenuse/Opposite |
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Cotangent |
Adjacent/Opposite |
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Pythagorean Identity |
Cos^2+Sin^2=1 |
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What direction are angles measured in? |
Counter-Clockwise |
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What direction are negative angles measured in? |
Clockwise |
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Coterminal angles are angles that... |
end at the same terminal side (405 is coterminal with 45) |
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Radians in a circle |
2π |
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To convert from radians to degrees, you multiply by... |
π/180 (cancel out the units then cross-multiply) |
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To convert form degrees to radians, you multiply by... |
180/π (cancel out the units then cross-multiply) |
