Functions & Patterns

What you’ll learn

• Graphing from a table is two jobs in order: plot every point exactly (across by $x$, then up or down by $y$), then step back and name the shape; five points is usually enough.
• For equal steps in $x$, the steps in $y$ name the family: constant steps make a line, same-size steps that flip direction make a V, steps that grow through the turn make a parabola, perfect-square $x$ values with slow $y$ growth make a square root, and rise-flatten-rise makes a cubic.
• Why must each allowable input of a function have only one output? If one input could pick two different answers, a table or graph would not describe a single clear rule; that is why graphs of $y=f(x)$ must pass the vertical line test.
• What is the most common reading mistake with $f(4)$? People sometimes treat the parentheses like multiplication; substitute $4$ for $x$ in the whole rule first, then simplify.
• When a table, equation, and graph describe the same story, what two anchors should you line up first? Check the $x=0$ row when it exists against the graph intercept, then check whether equal steps in $x$ always change $y$ by the same amount.
• How does proportional differ from a general linear model in a formula? Proportional means the line includes $(0,0)$ with $y=kx$; an affine model $y=mx+b$ allows a flat offset $b$ when the output at zero is not zero in context.