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AP Calculus Summer Enrichment Yay!

A bunch of stuff about math.
by

Sarah Timian

on 5 September 2012

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Transcript of AP Calculus Summer Enrichment Yay!

Linear Parent Graph Linear X and Y Intercepts The x intercept is 0 when y = 0.
The y intercept is 0 when x = 0. Domain and Range The domain of a linear equation is (-∞,∞).
The range is also (-∞,∞). Asymptotes Linear equations do not have asymptotes because they do not have denominators. Continuity Linear functions are continuous. Linear End Behavior As x -> (approaches) ∞, f(x) -> ∞. Even, Odd, or Neither Linear equations are odd because it is symmetrical when reflected about the x-axis and the y-axis. Intervals Linear equations are either always increasing or always decreasing. Number of Zeros On the x-axis, the max number of zeros is 1.
It is the same on the y-axis, only 1 zero. Extrema There are no extrema in a linear function. Quadratic Parent Graph Quadratic X and Y Intercepts One intercept at (0,0). Domain and Range The domain of a quadratic function is (-∞,∞).
The range is [0,∞). Asymptotes Like linear functions, quadratic functions don't have asymptotes because they lack a denominator. Continuity Quadratic functions are continuous. Quadratic End Behavior As x -> ∞, f(x) -> positive ∞. Even, Odd, or Neither Quadratic functions are even because they are symmetrical about the y-axis. Intervals Quadratic functions are decreasing from (-∞,0].
Quadratic functions are increasing from [0,∞). Extrema There is an absolute min. at (0,0). Cubic Parent Graph Cubic X and Y Intercepts The x and y intercept is (0,0) Domain and Range The domain of a cubic function is (-∞,∞).
The range of a cubic function is (-∞,∞) as well. Asymptotes Cubic functions do not have any asymptotes. Continuity Cubic functions are indeed continuous. End Behavior As x -> +∞, y-> +∞.
As x -> -∞, y -> -∞ as well. Cubic Even, Odd, or Neither Cubic functions are odd because the first quadrant is identical to the third quadrant when flipped over the y-axis, and the x-axis. Intervals Cubic functions are increasing from (-∞,∞). Number of Zeros There is one zero at (0,0) Extrema Cubic functions do not have any extrema. Square Root Parent Graph Square Root X and Y Intercepts The x intercept is at (0,0).
The y intercept is at (0,0) as well. Domain and Range The domain of a square root is [0,∞) The range is also [0,∞). Asymptotes Square root functions have no asymptotes. Continuity Square root functions are indeed continuous. End Behavior As x -> +∞, y -> +∞. Square Root Even, Odd, or Neither Neither, the square root function only exists in the first quadrant. Intervals The square root function is increasing from [0,∞). Number of Zeros Only one zero exists in the square root function at (0,0). Extrema The square root function has no extrema. Exponential Parent Graph Exponential X and Y Intercepts Exponential functions have no x intercepts.
They do have a y intercept at (0,1). Domain and Range The domain of the exponential function is (-∞,∞).
The range is (0,∞). Asymptotes Exponential functions have a horizontal asymptote at y = 0. Continuity Exponential functions are continuous. End Behavior As x -> -∞, y -> 0.
As x -> ∞, y -> ∞. Exponential Even, Odd, or Neither Neither, because the function is not symmetrical when reflected over neither the x-axis nor the y-axis. Intervals The exponential function is increasing from (-∞.∞). Number of Zeros There is only one zero in the exponential function at (0,1). Extrema Exponential functions have no extrema. Logarithmic Parent Graph Logarithmic X and Y Intercepts This function has an x intercept at (1,0).
Logarithmic functions do not have a y intercept. Domain and Range The domain of a logarithmic function is (0,∞).
The range is (-∞,∞). Asymptotes This function has a vertical asymptote at x = 0. Continuity The logarithmic function is continuous. End Behavior As x -> -∞, y does not equal 0. Therefore, x never reaches -∞.
As x -> ∞, y -> ∞. Logarithmic Even, Odd, or Neither Logarithmic functions are neither because they never exist reflected over the x or y-axis. Intervals This function is increasing from (0, ∞). Number of Zeros Logarithmic functions have only one zero at (1,0). Extrema There are no extrema in logarithmic functions. Sine Parent Graph Sine X and Y Intercepts Sine functions have the x and y intercept of (0,0).
They also have infinte x intercepts because of its repeating period. Domain and Range The domain of a sine functions is (-∞, ∞).
The range is (-1,1). Asymptotes This function does not have any asymptotes. Continuity This is a continuous function as well. End Behavior Sine functions have no "end behavior" because they are periodic functions. Sine Even, Odd, or Neither This function is odd because it is symmetrical when flipped over the x-axis and the y-axis. Intervals The sine function is increasing from (-π/2, π/2).
It is decreasing from (π/2, 3π/2).
This pattern repeats through each period. Number of Zeros The max number of zeros per period is three.
Throughout the entire function, there are an infinite amount of zeros. Extrema The sine function has an absolute min at (-π/2, -1) and an absolute max at (π/2, 1).
This repeats throughout each period. Period The period of a sine function is 2π. Cosine Parent Graph Cosine X and Y Intercepts The x intercept of this function is (π/2,0).
The y intercept is (0,1)
There are numerous intercepts after these because of the infinite periods in this function. Domain and Range The domain of the cosine function is (-∞,∞).
The range is [-1,1]. Asymptotes This function has no asymptotes. Continuity The cosine function is continuous. End Behavior The cosine function does not have any end behavior because it is a periodic function. Cosine Even, Odd, or Neither This function is even because it is symmetrical when reflected over the x-axis. Intervals This function is increasing from (-π/2, π/2) and decreasing from (π/2, 3π/2).
This continues throughout each period. Number of Zeros There is an infinite amount of zeros because it is a periodic function, but there is a max of three zeros per period. Extrema There is an absolute min of (π, -1) and an absolute max of (0,1).
This repeats every period. Tangent Parent Graph X and Y Intercepts The tangent function has an x intercept at π, and a y intercept at 0. (X intercept depends on the period). Domain and Range The domain is all real numbers except for π/2.
The range is all real numbers. Asymptotes. The tangent has one vertical asymptote at π/2. Continuity This function is not continous. End Behavior As x -> +∞, y -> -∞.
As x -> -∞, y -> +∞. Tangent Even, Odd, or Neither Intervals Number of Zeros Period The tangent function is odd because it is symmetrical which reflected over the x-axis, then the y-axis. The graph is increasing from -π/2 to π/2 in one period. Per period, the max number of zeros is one, but all together, the number is infinite. The period of the tangent function is π. Tangent Parent Graph Cosecant Cosecant X and Y Intercepts Domain and Range Asymptotes Continuity End Behavior Cosecant functions have no x or y intercepts. The domain is all real numbers except for π.
The range is (-∞, -1] U [1, ∞). This function has a vertical asymptote at π. This function is not continuous. The value of the sine repeats itself apart from sine in all four quadrants. Cosecant Even, Odd, or Neither Intervals Period Number of Zeros The cosecant function is odd because it is symmetrical when reflectded over the x and y-axis. The period of this function is 2π. Cosecant graphs have no zeros because they have no intercepts. This function is decreasing from (0, π/2) to (3π/2, 2π).
It is increasing from (π/2, π) to (π, 3π/2). Quartic Parent Graph Quartic Domain and Range The domain of this functions is (-∞, ∞).
The range is also (-∞, ∞). X and Y Intercepts The x intercepts are -3.9, -1.25, 1.25, and 2.8.
The y intercept is 1.4. Asymptotes The quartic function has no asymptotes. Continuity Yes, this function is continuous. End Behavior As x -> ∞, y -> ∞.
As x -> -∞, y -> ∞. Quartic Even, Odd, Or Neither The quartic function is neither because it is not symmetrical when flipped over the x or y-axis. Invervals This function is decreasing from -∞ to about 3.
It increases from about 3 to 1.4.
It then decreases from about 1.4 to 2.3.
Finally, it is increasing from about 2.3 to ∞. Number of Zeros This function has 4 zeros along the x-axis: -3.9, -1.25, 1.25, and 2.8. Extrema This function has no extrema. Cube Root Parent Graph Cube Root X and Y Intercepts The cube root function has an x and y intercept at (0,0). Domain and Range The domain is (-∞, ∞).
The range is (-∞, ∞) as well. Asymptotes The cube root function has no asymptotes. Continuity This function is continous. End Behavior As x -> ∞, y -> ∞.
As x -> -∞, y -> -∞. Cube Root Even, Odd, or Neither The cube root function is odd becaused it is symmetrical when reflected over the x and then y-axis. Intervals This function is increasing from -∞ to ∞. Number of Zeros The max number of zeros in the cube root function is one, at (0,0). Extrema This function has no extrema. Greatest Integer Parent Graph Greatest Integer X and Y Intercepts The x and y intercept of this function is (0,0). Domain and Range The domain of the greatest integer function is (-∞, ∞).
The range is also (-∞, ∞). Asymptotes This function have no asymptotes. Continuity The greatest integer function is very much not contiuous. End Behavior As x -> ∞, y -> ∞.
As x -> -∞, y -> -∞. Greatest Integer Even, Odd, or Neither This function is neither because it is not symmetrical when reflect over the x or y-axis. Intervals The greatest integer function is increasing from -∞ to ∞. Number of Zeros The max number of zeros in thie function is 1 at (0,0). Extrema This function has no extrema. Rational Function 1/x Parent Graph Rational Function 1/x X and Y Intercepts This function has no x and y intercepts because is has asymptotes at the x-axis and the y-axis. Domain and Range Both the domain and range are (-∞, 0)U(0,∞). Asymptotes In this function, there is an asymptote at the x-axis and at the y-axis. Continuity With both parts included, this graph is not contiuous. End Behavior As x -> ∞, y -> ∞.
As x -> -∞, y -> -∞. Rational Function 1/x Even, Odd, or Neither This function is odd because it is symmetrical when reflected over first the x and then the y-axis. Intervals The rational function 1/x is decreasing from -∞ to 0 (not included).
It is increasing from 0 (not included) to ∞. Number of Zeros This function has no zeros due to its asymptotes. Extrema This function has no extrema as well.
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