PSAT Advanced Math — shifting and reading nonlinear graphs

Many Advanced Math questions on the PSAT/NMSQT show a graph, a table or an equation of a nonlinear function and ask you to connect them — including, in the College Board's own skill descriptions, quadratic, exponential, polynomial and simple rational functions "that involve a transformation". This material trains the moves behind those questions: how a change to the equation moves the graph, and how to read the equation back from a shifted graph.

You learn the four basic moves and their signs. Adding k outside the function moves the graph up, subtracting moves it down; replacing x by x − h moves it h units to the right and x + h moves it to the left — the direction that surprises students most; a minus sign in front of the function reflects the graph across the x-axis. You follow single points through these moves, find the vertex of a shifted parabola, complete the square to see which translation turns y = x² into a given quadratic, and write the equation of a shifted absolute value graph from its vertex.

The same ideas apply to other families. You find the y-intercept of a shifted exponential function, the vertical asymptote of a shifted 1/x, and you decide from three points on a curve — using constant ratios rather than constant differences — whether the rule is exponential and what it is. A table question asks for g(2) when g(x) = f(x + 1), a common place to shift the output instead of the input.

Three questions come with a drawing on a coordinate grid: two parabolas of the same shape with marked vertices, an exponential curve through marked points, and a V-shaped graph with its vertex marked. The quiz has twelve multiple-choice questions whose explanations show where each wrong option comes from. The flashcards collect the transformation rules, vertices and asymptotes. The printable written work asks you to describe and justify transformations in your own words and to compute intercepts and asymptotes, as in student-produced responses. In the oral exam, an examiner gives you one function at a time and asks how its graph moves and why.

The content is based on the College Board's published skill descriptions for the PSAT/NMSQT Math section (Advanced Math: nonlinear functions). It is independent practice, not produced or endorsed by the College Board.

• Describe the graphs of f(x) + k, f(x − h) and −f(x) as shifts and reflections of the graph of f
• Follow a point of a graph through a transformation
• Find the vertex of a shifted parabola or absolute value graph and write its equation
• Find intercepts and asymptotes of shifted exponential and 1/x functions
• Match a nonlinear graph or table with its equation

Practice material written by Zestly, based on the College Board's skill descriptions for the digital PSAT/NMSQT Math section (Advanced Math: nonlinear functions — connections between tables, equations and graphs of quadratic, exponential, polynomial and simple rational functions, including those that involve a transformation), as published in the Assessment Framework for the Digital SAT Suite.