11-03: Various Types of Functions
Parent Functions
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Linear Function
f(x) = x
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Domain: {x E R}
Range: {y E R}
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Quadratic Function
f(x) = x²
| x | y |
|---|---|
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
Domain: {x E R}
Range: {y E R/0 ≤ y}
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Square Root Function
f(x) = √x
| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |
Domain: {x E R/0 ≤ x}
Range: {y E R/0 ≤ y}
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Reciprocal Function
f(x) = 1/x
| x | y |
|---|---|
| -2 | -1/2 |
| -1 | -1 |
| -0.5 | -2 |
| 0.5 | 2 |
| 1 | 1 |
| 2 | 1/2 |
Domain: {x E R/x ≠ 0}
Range: {y E R/y ≠ 0}
Asymptote: x = 0, y = 0
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Absolute Value Function
f(x) = |x|
| x | y |
|---|---|
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Domain: {x E R}
Range: {y E R/0 ≤ y}
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Cubic Function
f(x) = x³
| x | y |
|---|---|
| -2 | -8 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
Domain: {x E R}
Range: {y E R}
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Transformations of Parent Functions
Transformed functions: @@f(x) = a(k(x-d)) + c@@
- Vertical Stretch: ==a== * By a factor of…. * If negative, reflection in the x axis
- Horizontal Stretch: ^^k^^ * Always 1/k (flipped) * By a factor of…. * If negative, reflection in the y axis
- Vertical Translation: ==c== * if positive, moves up * If negative, moves down
- Horizontal Translation: ^^d^^ * Always the opposite sign of what it is in the brackets (sign is flipped) * If positive in bracket (so negative alone), then it moves left (
Mapping
- Draw the parent functions’ table of values
- Create mapping notation using:
Mapping Notation: @@((1/k)x + d, ay + c)@@
- Apply mapping notation to the parent function and graph (following BEDMAS, order of operations)
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Combinations of Transformations
| Quadratic | g(x) = a(k(x-d))² + c |
|---|---|
| Reciprocal | g(x) = a(1/(k(x-d)) + c |
| Cubic | g(x) = a(k(x-d))³ + c |
| Square Root | g(x) = a(√k(x-d) ) + c |
| Absolute Value | g(x) = a |k(x-d)| + c |
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Domain and Range of Functions from Equations
- Linear: * D: {x E R} * R: {y ER}
- Cubic: * D: {x E R} * R: {y ER}
- Quadratic: * D: {x E R} * R: {y E R/0 ≤ y} * c is the restriction (replacing zero)
- Absolute Value: * D: {x E R} * R: {y E R/0 ≤ y} * c is the restriction (replacing zero)
- Reciprocal: * D: {x E R/ x ≠ 0} * c replaces the restriction * R: {y E R/ y ≠ 0} * d replaces the restriction * The function cannot touch the asymptote thus the asymptote is our restriction
- Square Root: * D: {x E R / 0 ≤ x} * Make the number under the square root sign as small as it can be, so zero (because it cannot be negative since you can’t have a negative radicand) * R: {y E R / 0 ≤ y} * Look at what your lowest y could be as a result of the reduction of x for domain
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The Inverse Function
- Write the function in x-y notation
- Swap x and y
- Solve for y
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