Multiplication input/output tables: find the rule

Learning Objectives Analyze an input/output table to determine the constant multiplicative relationship representing a geometric dilation. Identify the scale factor 'k' for a dilation of a circle centered at the origin. Write the multiplication rule (e.g., r' = kr) that governs the relationship between an original circle's properties and its dilated image. Apply a determined multiplication rule to find the new coordinates of points on a dilated circle. Connect the algebraic concept of a multiplication rule to the geometric transformation of dilation. Use the multiplication rule to find the equation of a new circle after dilation. Ever wondered how animators make a cartoon sun grow bigger on screen? ☀️ They're using a mathematical 'rule' tha...

TermDefinitionExample Input/Output TableA table that organizes pairs of related numbers. The 'input' is the starting value (e.g., original radius), and the 'output' is the resulting value after applying a specific rule (e.g., new radius).If the rule is 'multiply by 4', an input of 3 gives an output of 12. Multiplication RuleA rule for an input/output table where the output is found by multiplying the input by a single, constant number. In geometry, this constant is the scale factor.Output = 5 × Input DilationA geometric transformation that changes the size of a figure but not its shape or orientation. It is performed with respect to a fixed point, called the center of dilation.Using a photocopier to enlarge a picture of a circle to 150% of its original size....

Finding the Rule (Scale Factor) k = \frac{\text{Output}}{\text{Input}} To find the constant multiplier 'k' in a multiplication input/output table, divide any output value by its corresponding input value. In the context of circles, this could be new radius / original radius. Dilation of a Point (x', y') = (kx, ky) To find the coordinates of a point on a dilated image (x', y'), multiply the original coordinates (x, y) by the scale factor 'k'. This rule applies to every point on the circle. Dilation of a Circle's Radius r' = kr The radius of the dilated circle (r') is found by multiplying the original radius (r) by the scale factor 'k'.