changes its corresponding graph on a Cartesian plane. We distinguish between transformations applied outside the function (affecting -coordinates) and inside the function (affecting -coordinates). . Affects the output. Inside change ( -axis): . Affects the input. Key Rule: Changes to behave normally (plus is up), while changes to behave inversely (plus is left). 2. Types of Graph Transformations (DSE Focus) A. Translations (Shifts) Vertical Shift: : Shift up : Shift down Horizontal Shift: : Shift right units (opposite of the sign!). : Shift left Example: Moving means shifting 2 units right and 3 units up. B. Reflections Reflection in -axis: The entire graph is flipped upside down. Points Reflection in -axis: The graph is flipped left-to-right. Points C. Stretches and Compressions Vertical Stretch/Compression: : Vertical stretch by factor : Vertical compression by factor Horizontal Stretch/Compression: : Horizontal compression by factor 1c1 over c end-fraction : Horizontal stretch by factor 1c1 over c end-fraction (opposite of the factor!). 3. Structure of a DSE Graph Transformation Exercise

We apply the transformations to the coordinates of the vertex.

| Function | Effect of (y = f(x-a) + b) | Effect of (y = k f(x)) | |----------|-------------------------------|--------------------------| | Quadratic (x^2) | Vertex shifts to (a, b) | Stretch in y-direction | | Exponential (e^x) | Horizontal shift = growth starting point change | Changes growth rate | | Logarithmic (\ln x) | Vertical shift changes horizontal asymptote? No, log has vertical asymptote at x = a after shift | Vertical stretch changes steepness | | Sine ( \sin x) | Horizontal shift = phase shift | Vertical stretch = amplitude change |

Should we focus on or trigonometric functions ( Share public link

Example 2: Graphical Matching (Paper 2 Multiple Choice Style) The figure shows the graph of . If the graph is transformed into , how does the new graph look compared to the old one? Solution:

This article will serve as a complete guide. We'll move from the fundamental rules of translations, reflections, and stretches to advanced problem-solving techniques. Most importantly, we'll bridge the gap between theory and practice with a dedicated section of DSE-style exercises, complete with detailed, step-by-step solutions.

At its heart, function transformation involves applying a mathematical operation to a known base function (like a quadratic or trigonometric function), which predictably alters its graph. The ultimate goal is often to determine the equation of a new graph or to visualize changes based on an equation.

-coordinates directly. The graph moves exactly as the sign suggests. shifts the graph upward by Vertical Translation Downward: shifts the graph downward by Vertical Stretching/Compressing: stretches the graph vertically by a factor of , and compresses it if Horizontal Transformations (Inside the Function)

Find the equation of the new graph. Then find the domain and range.

horizontally , and then translating it vertically upward by 5 units . To find the new vertex P′cap P prime -coordinate: -coordinate: Answer: Question 2 . The graph of undergoes the following consecutive transformations: Reflected in the Compressed vertically by a factor of 12one-half Translated downward by 3 units. Find the final equation of the transformed graph, , expressing your answer in the form . (4 marks) Solution: Step 1 (Reflection in y-axis): Replace Step 2 (Vertical compression by 12one-half ): Multiply the entire function by 12one-half