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Basic Graphing
Point plotting is plotting points until the shape of the function is apparent.
If you don’t plot enough points you risk misinterpreting the graph.
Intercepts are points that have 0 as either the X or Y coordinates.
X-intercepts are points that lie on the x-axis, so Y coordinate is 0.
Y-intercepts are points that lie on the y-axis, so X coordinate is 0.
Knowing the symmetry of a graph can cut the amount of points you need by half.
Symmetry around the y-axis :
f ( x ) = f ( − x ) f(x) = f(-x) f ( x ) = f ( − x ) Also called even functions.
Symmetry around the x-axis :
f ( x ) = − f ( x ) f(x) = -f(x) f ( x ) = − f ( x ) Also called odd functions.
Symmetry around the origin :
f ( x ) = − f ( − x ) f(x) = -f(-x) f ( x ) = − f ( − x )
In order to test for symmetry, replace your function’s X and/or Y with the corresponding -X and/or -Y for the symmetry you are checking and see if the functions are equal. If they are equal, it is symmetrical around that axis.
Find X and Y intersections
Test symmetries
Plot points
Domain - Inputs(x)Range - Outputs(y)
Name
Function
Image
Name
Function
Image
Line
y = x y = x y = x Squared
y = x 2 y = x^2 y = x 2
Cubed
y = x 3 y = x^3 y = x 3 Square root
y = x y = \sqrt{x} y = x
Abs
y = x y = x y = x Rational
y = 1 x y = \frac{1}{x} y = x 1
s l o p e = m = r i s e r u n = y 1 − y 2 x 1 − x 2 = y 2 − y 1 x 2 − x 1 slope = m = \frac{rise}{run} = \frac{y_1 - y_2}{x_1 - x_2} = \frac{y_2 - y_1}{x_2 - x_1} s l o p e = m = r u n r i se = x 1 − x 2 y 1 − y 2 = x 2 − x 1 y 2 − y 1
Positive slope: /, negative slope: \, zero slope: _, undefined slope: |
Form Name
Equation
Point Slope
y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) y − y 1 = m ( x − x 1 )
Slope Intercept
y = m x + b y = mx + b y = m x + b
General
a x + b y + c = 0 ax + by + c= 0 a x + b y + c = 0
Vertical line
x = a
Horizontal line
y = a
Parallel - two lines have the same slope. m 1 = m 2 m_1 = m_2 m 1 = m 2
Perpendicular - two lines have negative reciprocals as slopes. m 1 = − 1 m 2 m_1 = -\frac{1}{m_2} m 1 = − m 2 1
For the slope intercept form, the Y-intercept is (0, b).
f ( x ) = a n x n + a n − 1 x n − 1 + … + a 2 x 2 + a 1 x 1 + a 0 f(x) = a_{n}x^n + a_{n-1}x^{n-1} + … + a_{2}x^2 + a_{1}x^1 + a_0 f ( x ) = a n x n + a n − 1 x n − 1 + … + a 2 x 2 + a 1 x 1 + a 0
n n n a a a a n a_n a n a 0 a_0 a 0
Even degree polynomials:
a n > 0 a_n > 0 a n > 0 a n < 0 a_n < 0 a n < 0
Odd degree polynomials:
a n > 0 a_n > 0 a n > 0 a n < 0 a_n < 0 a n < 0
Name
Function
Example
Shift left
f ( x + c ) f(x + c) f ( x + c ) y = ( x + 2 ) 2 y = (x + 2)^2 y = ( x + 2 ) 2
Shift right
f ( x − c ) f(x - c) f ( x − c ) y = ( x − 2 ) 2 y = (x - 2)^2 y = ( x − 2 ) 2
Shift up
f ( x ) + c f(x) + c f ( x ) + c y = x 2 + 2 y = x^2 + 2 y = x 2 + 2
Shift down
f ( x ) − c f(x) - c f ( x ) − c y = x 2 − 2 y = x^2 - 2 y = x 2 − 2
Reflect y-axis
f ( − x ) f(-x) f ( − x ) y = ( − x ) 2 = x 2 y = (-x)^2 = x^2 y = ( − x ) 2 = x 2
Reflect x-axis
− f ( x ) -f(x) − f ( x ) y = − x 2 y = -x^2 y = − x 2
Reflect origin
− f ( − x ) -f(-x) − f ( − x ) y = − ( − x ) 2 = − x 2 y = -(-x)^2 = -x^2 y = − ( − x ) 2 = − x 2
Squeeze x-axis
f ( a x ) f(ax) f ( a x ) y = ( a x ) 2 y = (ax)^2 y = ( a x ) 2
Stretch x-axis
f ( x a ) f(\frac{x}{a}) f ( a x ) y = ( x a ) 2 y = (\frac{x}{a})^2 y = ( a x ) 2
Elementary functions are functions that can represent many real world phenomena.
Algebraic functions
Trigonometric functions
Exponential and logarithmic functions
Algebraic functions are functions that can combine multiple algebraic operations on polynomials.
Operation name
Function Shorthand
Function Extended
Sum
( f + g ) ( x ) (f + g)(x) ( f + g ) ( x ) f ( x ) + g ( x ) f(x) + g(x) f ( x ) + g ( x )
Difference
( f − g ) ( x ) (f - g)(x) ( f − g ) ( x ) f ( x ) − g ( x ) f(x) - g(x) f ( x ) − g ( x )
Product
( f g ) ( x ) (fg)(x) ( f g ) ( x ) f ( x ) ∗ g ( x ) f(x) * g(x) f ( x ) ∗ g ( x )
Quotient
( f / g ) ( x ) (f/g)(x) ( f / g ) ( x ) f ( x ) / g ( x ) f(x) / g(x) f ( x ) / g ( x )
Composition
( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) f ( g ( x ) ) f(g(x)) f ( g ( x ))
Radical functions are the same as square root: f ( x ) = g ( x ) n f(x) = \sqrt[n]{g(x)} f ( x ) = n g ( x )
A point that satisfies both equations.
Make sure both equations equal Y
Set both equations equal to each other
Solve for X
Find the corresponding Y coordinates for the X coordinates.
