Transformations of Functions
1. General Translations
Horizontal Translations: To translate a graph horizontally, adjust the x-value by adding (shift left) or subtracting (shift right)
Vertical Translations: For vertical translations, adjust the y-value by adding (shift up) or subtracting (shift down)
2. Transformations Involving Stretches
If k > 1, the graph stretches vertically
If 0 < k < 1, it compresses vertically.
Apply this factor to the function's output values (y).
3. Combining Transformations
Apply the stretch/compression (k: (x,ky)) first
Apply horizontal & vertical translations (h: h±x,y))
5. Reflections
Horizontal reflection: f(-x) reflect the graph over y-axis (-x,y)
Vertical reflection: f(x) = -f(x) reflect the graph over the x-axis (x,-y)
6. Characteristics of Polynomial Functions
Pay attention to the degree (highest exponent) and leading coefficient to evaluate end behavior. Roots can be found by factoring or using synthetic division.
7. Factor Theorem and Roots
Use synthetic division to test for potential roots. A number is a root if substituting it into the polynomial yields zero, and you can factor the polynomial accordingly.
8. Rational Functions
Analyze asymptotes: vertical asymptotes occur when the denominator equals zero (undefined), while horizontal asymptotes depend on degrees of numerator and denominator.
9. Exponential and Logarithmic Functions
Base determines whether the function:
exponential f(x)= a x b^x
logarithmic f(x) = logb(x)
grows: base is greater than 1
decays: base is between 0 and 1
10. Trigonometric Functions
amplitude: height of the wave from the midline (coefficient in front of sine or cosine)
period: length of one complete cycle
phased shift: horizontal shift (determined by adding/subtracting to the x-variable inside the function
11. Combinations and Permutations
Apply combinations when order does not matter (C(n, r) = n! / [r!(n - r)!]).
Use permutations when order does matter (P(n, r) = n! / (n - r)!).
12. Graphing Real World Applications
Use the functions to model scenarios, identifying periodic behavior (like seasons or trends) and recognizing how to apply the transformations to fit data.
Polynomial Functions
Linear (degree 1)
domain: xER
range: yER
end behavior: f(x)—> infinity a>0 / f(x)—> -infinity a<0
0=ax+b
x intercept: -b/a
y intercept: b
Quadratic (degree 2)
domain: xER
range: yER
end behavior: f(x)—> infinity a>0 / f(x)—> -infinity a>0
ax² + bx +c =0
x intercept: use quadratic formula
y intercept: f(0)=c
Cubic (degree 3)
domain: xER
range: yER
end behavior: f(x)—> infinity a>0 / f(x)—> -infinity a>0
ax³ + bx² + cx + d
x intercept: set f(x)=0 and solve (can have 1,2,3)
y intercept: f(0)=d
Quartic (degree 4)
domain: xER
range: a>0 has minimum value, a<0 has maximum value
end behavior: f(x)—> infinity a>0 / f(x)—> -infinity a>0
ax^4+ bx³ + cx² + dx + e
x intercept: f(x)=0 (can have up to 4)
y intercept f(x)=e
Quintic (degree 5)
domain: xER
range: yER
end behavior: f(x)—> infinity a>0 / f(x)—> -infinity a>0
ax^5 + bx^4 + cx³ + dx² + ex + f
x intercepts: f(x)=0 (can be up to 5)
y intercepts: f(0)=f
Radical Functions
f(x)= √g(x)
domain: g(x)>0 (cant square root negative numbers) - take what is under the square root and solve for x
range: set x=0 and solve for y
Rational Functions
f(x)= p(x)/q(x) - q(x) cannot be zero
domain: set denominator to 0 and include NPVs
range: depends
vertical asymptotes: NPVs from domain
horizontal asymptotes:
leading degree of coefficients are same (numerator/denominator)
leading degree of coefficients numerator is greater than denominator = none
leading degree of coefficients numerator is less than denominator = 0
x intercept: set numerator to 0 and solve for x
y intercept: set x to 0 solve for x