Radical Functions and Equations
Lessons in this chapter
About Radical Functions and Equations
Radical Functions and Equations covers the two most common radical function families and how to solve equations involving them.
Graphing Square Root Functions and Graphing Cube Root Functions cover the distinctive graphs of each function family: a square root function's graph starts at a point and curves in one direction, since negative inputs aren't allowed, while a cube root function's graph is defined for every real input and has a distinctive S-curve shape.
Solving Radical Equations closes the chapter by isolating the radical term and raising both sides to a matching power to eliminate it: a step that can introduce extraneous solutions, so checking every answer back in the original equation is a required last step, not an optional one.
This chapter has three lessons and each opens as an Algebra 1 worksheet in the page: Graphing Square Root Functions, Graphing Cube Root Functions and Solving Radical Equations. The questions are scored as you answer them, which is particularly useful on the last lesson, where squaring both sides can produce a solution that does not actually work in the original equation. The worksheets are free to try, with no login needed for your first 3.
Frequently asked questions
Graphing Square Root Functions introduces a graph that starts at a point and curves away in one direction only, because a square root is undefined for negative inputs, and Graphing Cube Root Functions contrasts it with a curve defined for every input. Solving Radical Equations then isolates the radical and raises both sides to the matching power, with the important habit of checking each answer back in the original equation, since squaring can introduce solutions that do not hold. The chapter draws on the high school Functions (F) and Algebra (A) conceptual categories.
A square root of a negative number isn't a real number, so a square root function's domain is restricted to inputs that keep the expression under the radical non-negative: its graph starts at a point and extends in only one direction.
A cube root of a negative number IS a real number (the cube root of -8 is -2, since -2 × -2 × -2 = -8), so a cube root function is defined for every real input, giving it a full S-curve rather than a graph that starts at a point.
Isolate the radical term on one side, then raise both sides of the equation to a matching power (square both sides for a square root, cube both sides for a cube root) to eliminate the radical.
An extraneous solution is a value that comes out of the solving process but doesn't actually satisfy the original equation. It happens because raising both sides to a power can introduce solutions that weren't valid in the first place, so every answer must be checked in the original equation.
Skipping the final check, since raising both sides to a power can introduce extraneous solutions, every candidate solution must be substituted back into the ORIGINAL radical equation to confirm it actually works.