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Pre-Calculus Function Transformations cheat sheet - grade 9-12

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Math Grade 9-12

Pre-Calculus Function Transformations Cheat Sheet

A printable reference covering parent functions, shifts, reflections, stretches, compressions, and transformation order for grades 9-12.

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Function transformations show how a graph changes when numbers are added, subtracted, multiplied, or placed inside a function rule. This cheat sheet helps students recognize shifts, reflections, stretches, and compressions without graphing every point. It is useful for comparing transformed functions to parent functions in algebra, pre-calculus, and graphing problems.

Understanding transformations makes it easier to write equations from graphs and predict graph behavior quickly.

The core idea is to compare a transformed function such as g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k to its parent function f(x)f(x). The value hh moves the graph horizontally, kk moves it vertically, aa changes vertical size and reflection, and bb changes horizontal size and reflection. Transformations inside the input affect the graph horizontally and often work in the opposite direction students expect.

Transformations outside the function affect the graph vertically and usually work in the expected direction.

Key Facts

  • The general transformation form is g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k, where aa, bb, hh, and kk control the graph.
  • The graph of g(x)=f(x)+kg(x) = f(x) + k shifts the parent graph vertically up kk units if k>0k > 0 and down ∣k∣|k| units if k<0k < 0.
  • The graph of g(x)=f(x−h)g(x) = f(x - h) shifts the parent graph right hh units if h>0h > 0 and left ∣h∣|h| units if h<0h < 0.
  • The graph of g(x)=af(x)g(x) = a f(x) is vertically stretched by factor ∣a∣|a| if ∣a∣>1|a| > 1 and vertically compressed by factor ∣a∣|a| if 0<∣a∣<10 < |a| < 1.
  • If a<0a < 0 in g(x)=af(x)g(x) = a f(x), the graph is reflected across the xx-axis.
  • The graph of g(x)=f(bx)g(x) = f(bx) is horizontally compressed by factor 1∣b∣\frac{1}{|b|} if ∣b∣>1|b| > 1 and horizontally stretched by factor 1∣b∣\frac{1}{|b|} if 0<∣b∣<10 < |b| < 1.
  • If b<0b < 0 in g(x)=f(bx)g(x) = f(bx), the graph is reflected across the yy-axis.
  • For g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k, points on f(x)f(x) transform from (x,y)(x,y) to (h+xb,ay+k)(h + \frac{x}{b}, ay + k).

Vocabulary

Parent function
A parent function is the simplest function in a family, such as f(x)=x2f(x) = x^2, f(x)=∣x∣f(x) = |x|, or f(x)=xf(x) = \sqrt{x}.
Translation
A translation slides a graph horizontally or vertically without changing its shape.
Reflection
A reflection flips a graph across a line, such as the xx-axis for g(x)=−f(x)g(x) = -f(x) or the yy-axis for g(x)=f(−x)g(x) = f(-x).
Vertical stretch or compression
A vertical stretch or compression multiplies all output values by aa in g(x)=af(x)g(x) = a f(x).
Horizontal stretch or compression
A horizontal stretch or compression changes input values by bb in g(x)=f(bx)g(x) = f(bx), using the scale factor 1∣b∣\frac{1}{|b|}.
Transformation form
Transformation form is g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k, which organizes vertical changes, horizontal changes, and reflections.

Common Mistakes to Avoid

  • Treating f(x−h)f(x - h) as a shift left is wrong because changes inside the input work oppositely; f(x−h)f(x - h) shifts the graph right hh units when h>0h > 0.
  • Forgetting the reciprocal in horizontal scaling is wrong because g(x)=f(bx)g(x) = f(bx) changes widths by 1∣b∣\frac{1}{|b|}, not by ∣b∣|b|.
  • Confusing −f(x)-f(x) and f(−x)f(-x) is wrong because −f(x)-f(x) reflects across the xx-axis, while f(−x)f(-x) reflects across the yy-axis.
  • Applying transformations in a random order can give the wrong equation or graph; use g(x)=af(b(x−h))+kg(x) = a f(b(x - h)) + k to identify horizontal changes, then vertical changes.
  • Ignoring parentheses in expressions like f(2(x−3))f(2(x - 3)) is wrong because the horizontal shift is 33 units right, not 66 units right.

Practice Questions

  1. 1 Describe all transformations from f(x)=x2f(x) = x^2 to g(x)=−2(x−3)2+5g(x) = -2(x - 3)^2 + 5.
  2. 2 If the point (4,7)(4, 7) is on f(x)f(x), find the corresponding point on g(x)=3f(2(x−1))−4g(x) = 3f(2(x - 1)) - 4.
  3. 3 Write a transformed function based on f(x)=xf(x) = \sqrt{x} that shifts left 22 units, reflects across the xx-axis, and shifts up 66 units.
  4. 4 Explain why g(x)=f(x+4)g(x) = f(x + 4) moves the graph left instead of right, even though 44 is positive.

Understanding Pre-Calculus Function Transformations

A useful way to work with transformations is to track a few important points instead of making a large table. For a parabola, the vertex is usually the best starting point. For an absolute value graph, use the corner.

For a square root graph, use its endpoint. For an exponential graph, notice the horizontal asymptote, which is a line the graph approaches without reaching. A point on the original graph has an input and an output.

The transformed rule changes both parts of that point. Vertical changes alter outputs. Horizontal changes alter inputs.

This point mapping explains why a horizontal scale factor behaves like a reciprocal. If inputs are multiplied by a number greater than one inside the rule, each original x-value must be reached sooner, so the picture becomes narrower.

The order of operations matters because transformations are combined rules, not separate decorations. Begin with the expression closest to the input. Grouping symbols tell you what belongs together.

A horizontal shift written inside a factor must be read as one complete input rule. For example, a graph based on the input two times the quantity x minus three does not simply move right three units and become narrower. To find its horizontal location carefully, set the complete input equal to the original input value.

This method prevents most sign mistakes. It is especially helpful when a graph has both a reflection and a shift. A reflection can reverse the direction of features, while a later shift moves the whole reflected graph.

Different parent functions keep recognizable features after transformation. A line remains a line, though its slope and intercept can change. A quadratic remains a parabola, with its opening direction, vertex, and axis of symmetry giving strong clues.

An absolute value graph keeps its sharp corner. A cubic keeps its central turning shape. For rational functions, vertical and horizontal asymptotes move or change according to the rule.

These features are often more reliable than individual plotted points. They help when a graph is partly off the screen or has a scale that is not one unit per grid square. Pay attention to the domain and range too.

A square root graph may begin at a new endpoint. A logarithmic graph may have a shifted vertical boundary where inputs are not allowed.

Transformations appear whenever one relationship is adjusted to fit a situation. A temperature conversion shifts a linear rule and changes its scale. A sound waveform can be made taller, quieter, delayed, or reversed.

In science classes, measured data may follow the same curve as a model but use different units, starting values, or time scales. When learning, sketch the parent graph lightly first, then mark its key features after each change. Check one or two points by substituting their inputs into the new rule.

Finally, use a graphing tool as a check rather than as the only method. The goal is to connect the equation, the moved points, and the visible shape.