Functions and Proportions - Math 1 & 2 PDF

**Functions** A function is a relationship between two sets of numbers or variables where each input (usually ( x )) is related to exactly one output (usually ( y )). For example, in the function ( y = 2x + 3 ), if you input a value for ( x ), you get a specific value for ( y ). **Proportion** ｆ...

**Functions** A function is a relationship between two sets of numbers or variables where each input (usually ( x )) is related to exactly one output (usually ( y )). For example, in the function ( y = 2x + 3 ), if you input a value for ( x ), you get a specific value for ( y ). **Proportion** ｆProportion refers to the relationship between two quantities where their ratio remains constant. For example, if ( y ) is directly proportional to ( x ), it can be written as ( y = kx ), where ( k ) is the constant of proportionality. If ( x ) doubles, ( y ) also doubles. **Inverse Proportion** Inverse proportion describes a relationship where one quantity increases as the other decreases. If ( y ) is inversely proportional to ( x ), it can be written as ( y = \\frac{k}{x} ), where ( k ) is a constant. For example, if ( x ) doubles, ( y ) is halved. **Applying Proportion and Inverse Proportion** - **Direct Proportion**: Used in situations where quantities increase or decrease together. For example, if you buy more apples, the total cost increases proportionally. - **Inverse Proportion**: Applied in scenarios where one quantity decreases as another increases. For example, the speed of a car and the time taken to travel a fixed distance are inversely proportional. **Inequalities** are mathematical expressions that show the relationship between two values where they are not equal. They use symbols like (\>) (greater than), (\ 3 ) is shown as an open circle at 3 with a line extending to the right. - **Compound Inequalities**: Combine two inequalities using \"and\" or \"or\". For example, ( 1 \< x \< 5 ) means ( x ) is between 1 and 5. **Functions and Proportions (Math 1 Textbook, pages 98 - 118, Shin Chu Mon 7, pages 95 - 114)** **Functions**: - A function is a relationship where each input (usually ( x )) has exactly one output (usually ( y )). - **Linear Functions**: Represented by ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. - **Graphing Functions**: Plot points on a coordinate plane and draw the line or curve that represents the function. **Proportions**: - **Direct Proportion**: Two quantities increase or decrease together at the same rate, represented by ( y = kx ) where ( k ) is the constant of proportionality. - **Inverse Proportion**: One quantity increases as the other decreases, represented by ( y = \\frac{k}{x} ). **Calculating with Expressions / Using Expressions (Math 2 Textbook, pages 13-26, Shin Chu Mon 8, pages 7-15, 17-22)** **Calculating with Expressions**: - **Simplifying Expressions**: Combine like terms and use the distributive property to simplify algebraic expressions. - **Evaluating Expressions**: Substitute values for variables and perform the arithmetic operations. **Using Expressions**: - **Formulas**: Use algebraic expressions to represent real-world situations. For example, the area of a rectangle ( A = lw ) where ( l ) is length and ( w ) is width. - **Solving Equations**: Set expressions equal to each other and solve for the unknown variable. Proportional: y=ax a=x/y x=y/a Inverse Proportional: y=a/x a=xy x=a/y