Graphs of other Functions
Graphs of other Functions Miles Honey
Standard functions • y=x2
• Y=x3
•Y=x4
•Y=2x2-12x+7
Reciprocal functions •
Y=x1/2
•Y=x1/5
•Y=x3/2
•Y=x5/4
Features of reciprocal functions • Almost a parabola on it’s side • Functions with even denominators only stay in positive quadrant • Top-heavy fractions change slant to normal ones. • All pass through (0,0)
Surd functions • Y=xsqrt(x)
•Y=xsqrt(2)
•Y=xsqrt(3)
Features of surd functions • Only in the positive quadrant- no square root of a negative number • Not all pass through (0,0) • All have positive, increasing gradients
Surd AND reciprocal functions • Y=xsqrt(1/2)
•Y=xsqrt(3/2)
•Y=xsqrt(2/3)
•Y=x sqrt(5/4)
• •
Graph generator can be found at: http://www.analyzemath.com/graphing_calculators/graphing_inverse.html
Graphs
y=x²
y=x^-2
Y=x^1/2 or y=root ofx
Tom Reading
y=x³
y=x^-3
Reciprocal Graphs
If you have a graph of y=k/x², where k is positive, then the graph will always be above the x-axis This is because any number squared is positive, so your value for y will always be positive even if x is negative If k is negative however, y will always be negative, meaning the graph will always be under the x-axis This type of reciprocal graphs always have one line of symmetry at x=0
This is a graph of the function y=5/x, a basic reciprocal graph As the graph gets closer to the x-axis as the value of x increases, but it never meets the axis. Each piece of the graph also gets closer to the y-axis as x gets closer to 0 but it never meets the y-axis because there is no value for y when x = 0. This type of curve is called a rectangular hyperbola If the graph to the left was y=5/(-x), then you would simply reflect the graph on the x-axis Reciprocal functions which take the form y=k/x always have 2 lines of symmetry at y=x and y=x, as shown in the diagram below
Sketching Graphs By Michael Stamps
Linear Y=-2x+4 When Y=0 x=4
quadratic Y=-4x^2+2x+6 When Y=0 x=6 Y=x^2+5 Y=x^2-10
Y=2x^2+5 Y=5x^2
Cubic Alternative graph
Y=x^3+5 Y=-2x^3+10 Y=x^3
Reciprocal
Y=-a/X
examples
Standard functions • y=x2
• Y=x3
•Y=x4
•Y=2x2-12x+7
Reciprocal functions •
Y=x1/2
•Y=x1/5
•Y=x3/2
•Y=x5/4
Features of reciprocal functions • Almost a parabola on it’s side • Functions with even denominators only stay in positive quadrant • Top-heavy fractions change slant to normal ones. • All pass through (0,0)
Surd functions • Y=xsqrt(x)
•Y=xsqrt(2)
•Y=xsqrt(3)
Features of surd functions • Only in the positive quadrant- no square root of a negative number • Not all pass through (0,0) • All have positive, increasing gradients
Surd AND reciprocal functions • Y=xsqrt(1/2)
•Y=xsqrt(3/2)
•Y=xsqrt(2/3)
•Y=x sqrt(5/4)
• •
Graph generator can be found at: http://www.analyzemath.com/graphing_calculators/graphing_inverse.html
Graphs
y=x²
y=x^-2
Y=x^1/2 or y=root ofx
Tom Reading
y=x³
y=x^-3
Reciprocal Graphs
If you have a graph of y=k/x², where k is positive, then the graph will always be above the x-axis This is because any number squared is positive, so your value for y will always be positive even if x is negative If k is negative however, y will always be negative, meaning the graph will always be under the x-axis This type of reciprocal graphs always have one line of symmetry at x=0
This is a graph of the function y=5/x, a basic reciprocal graph As the graph gets closer to the x-axis as the value of x increases, but it never meets the axis. Each piece of the graph also gets closer to the y-axis as x gets closer to 0 but it never meets the y-axis because there is no value for y when x = 0. This type of curve is called a rectangular hyperbola If the graph to the left was y=5/(-x), then you would simply reflect the graph on the x-axis Reciprocal functions which take the form y=k/x always have 2 lines of symmetry at y=x and y=x, as shown in the diagram below
Sketching Graphs By Michael Stamps
Linear Y=-2x+4 When Y=0 x=4
quadratic Y=-4x^2+2x+6 When Y=0 x=6 Y=x^2+5 Y=x^2-10
Y=2x^2+5 Y=5x^2
Cubic Alternative graph
Y=x^3+5 Y=-2x^3+10 Y=x^3
Reciprocal
Y=-a/X
examples
