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Study Guides > MATH 1314: College Algebra

Key Concepts & Glossary

Key Equations

Vertical shift g(x)=f(x)+kg\left(x\right)=f\left(x\right)+k\\ (up for k>0k>0\\ )
Horizontal shift g(x)=f(xh)g\left(x\right)=f\left(x-h\right)\\ (right for h>0h>0\\ )
Vertical reflection g(x)=f(x)g\left(x\right)=-f\left(x\right)\\
Horizontal reflection g(x)=f(x)g\left(x\right)=f\left(-x\right)\\
Vertical stretch g(x)=af(x)g\left(x\right)=af\left(x\right)\\ ( a>0a>0\\)
Vertical compression g(x)=af(x)g\left(x\right)=af\left(x\right)\\ (0<a<1)\left(0<a<1\right)\\
Horizontal stretch g(x)=f(bx)g\left(x\right)=f\left(bx\right)\\ (0<b<1)\left(0<b<1\right)\\
Horizontal compression g(x)=f(bx)g\left(x\right)=f\left(bx\right)\\ ( b>1b>1\\ )

Key Concepts

  • A function can be shifted vertically by adding a constant to the output.
  • A function can be shifted horizontally by adding a constant to the input.
  • Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.
  • Vertical and horizontal shifts are often combined.
  • A vertical reflection reflects a graph about the x-x\text{-}\\ axis. A graph can be reflected vertically by multiplying the output by –1.
  • A horizontal reflection reflects a graph about the y-y\text{-}\\ axis. A graph can be reflected horizontally by multiplying the input by –1.
  • A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.
  • A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.
  • A function presented as an equation can be reflected by applying transformations one at a time.
  • Even functions are symmetric about the y-y\text{-}\\ axis, whereas odd functions are symmetric about the origin.
  • Even functions satisfy the condition f(x)=f(x)f\left(x\right)=f\left(-x\right)\\.
  • Odd functions satisfy the condition f(x)=f(x)f\left(x\right)=-f\left(-x\right)\\.
  • A function can be odd, even, or neither.
  • A function can be compressed or stretched vertically by multiplying the output by a constant.
  • A function can be compressed or stretched horizontally by multiplying the input by a constant.
  • The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order.

Glossary

even function
a function whose graph is unchanged by horizontal reflection, f(x)=f(x)f\left(x\right)=f\left(-x\right)\\, and is symmetric about the y-y\text{-}\\ axis
horizontal compression
a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant b>1b>1\\
horizontal reflection
a transformation that reflects a function’s graph across the y-axis by multiplying the input by 1-1\\
horizontal shift
a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
horizontal stretch
a transformation that stretches a function’s graph horizontally by multiplying the input by a constant 0<b<10<b<1\\
odd function
a function whose graph is unchanged by combined horizontal and vertical reflection, f(x)=f(x)f\left(x\right)=-f\left(-x\right)\\, and is symmetric about the origin
vertical compression
a function transformation that compresses the function’s graph vertically by multiplying the output by a constant 0<a<10<a<1\\
vertical reflection
a transformation that reflects a function’s graph across the x-axis by multiplying the output by 1-1\\
vertical shift
a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
vertical stretch
a transformation that stretches a function’s graph vertically by multiplying the output by a constant a>1a>1\\

Licenses & Attributions